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== 5.4#h(0.6em)Dividing Polynomials

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Add: #math.equation(block: false, alt: "the fraction 3 over d plus the fraction x over d .")[$frac(3, d) + frac(x, d) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 plus x over d")[$frac(3 + x, d)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 30 x y cubed over 5 x y .")[$frac(30 x y^(3), 5 x y) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "6 y squared")[$6 y^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Combine like terms: #math.equation(block: false, alt: "8 a squared plus 12 a plus 1 plus 3 a squared minus 5 a plus 4 .")[$8 a^(2) + 12 a + 1 + 3 a^(2) − 5 a + 4 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "11 a squared plus 7 a plus 5")[$11 a^(2) + 7 a + 5$]
]
]


=== Dividing Monomials

We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials.

#examplebox("Example 1")[][
Find the quotient: #math.equation(block: false, alt: "54 a squared b cubed divided by open parenthesis −6 a b to the power 5 close parenthesis .")[$54 a^(2) b^(3) ÷ ( −6 a b^(5) ) .$]

#solutionbox[
When we divide monomials with more than one variable, we write one fraction for each variable. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "54 a squared b cubed divided by open parenthesis −6 a b to the power 5 close parenthesis")[$54 a^(2) b^(3) ÷ ( −6 a b^(5) )$]]),
  [Rewrite as a fraction.], [#math.equation(block: false, alt: "the fraction 54 a squared b cubed over −6 a b to the power 5")[$frac(54 a^(2) b^(3), −6 a b^(5))$]],
  [Use fraction multiplication.], [#math.equation(block: false, alt: "the fraction 54 over −6 times the fraction a squared over a times the fraction b cubed over b to the power 5")[$frac(54, −6) · frac(a^(2), a) · frac(b^(3), b^(5))$]],
  [Simplify and use the Quotient Property.], [#math.equation(block: false, alt: "−9 times a times the fraction 1 over b squared")[$−9 · a · frac(1, b^(2))$]],
  [Multiply.], [#math.equation(block: false, alt: "minus the fraction 9 a over b squared")[$− frac(9 a, b^(2))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "−72 a to the power 7 b cubed divided by open parenthesis 8 a to the power 12 b to the power 4 close parenthesis .")[$−72 a^(7) b^(3) ÷ ( 8 a^(12) b^(4) ) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 9 over a to the power 5 b")[$− frac(9, a^(5) b)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "−63 c to the power 8 d cubed divided by open parenthesis 7 c to the power 12 d squared close parenthesis .")[$−63 c^(8) d^(3) ÷ ( 7 c^(12) d^(2) ) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction −9 d over c to the power 4")[$frac(−9 d, c^(4))$]
]
]

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

#examplebox("Example 2")[][
Find the quotient: #math.equation(block: false, alt: "the fraction 14 x to the power 7 y to the power 12 over 21 x to the power 11 y to the power 6 .")[$frac(14 x^(7) y^(12), 21 x^(11) y^(6)) .$]

#solutionbox[
Be very careful to simplify #math.equation(block: false, alt: "the fraction 14 over 21")[$frac(14, 21)$] by dividing out a common factor, and to simplify the variables by subtracting their exponents. #linebreak()

#figure(table(
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  table.header([], [#math.equation(block: false, alt: "the fraction 14 x to the power 7 y to the power 12 over 21 x to the power 11 y to the power 6")[$frac(14 x^(7) y^(12), 21 x^(11) y^(6))$]]),
  [Simplify and use the Quotient Property.], [#math.equation(block: false, alt: "the fraction 2 y to the power 6 over 3 x to the power 4")[$frac(2 y^(6), 3 x^(4))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 28 x to the power 5 y to the power 14 over 49 x to the power 9 y to the power 12 .")[$frac(28 x^(5) y^(14), 49 x^(9) y^(12)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 y squared over 7 x to the power 4")[$frac(4 y^(2), 7 x^(4))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 30 m to the power 5 n to the power 11 over 48 m to the power 10 n to the power 14 .")[$frac(30 m^(5) n^(11), 48 m^(10) n^(14)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 8 m to the power 5 n cubed")[$frac(5, 8 m^(5) n^(3))$]
]
]


=== Divide a Polynomial by a Monomial

Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.

The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition. The sum #math.equation(block: false, alt: "the fraction y over 5 plus the fraction 2 over 5")[$frac(y, 5) + frac(2, 5)$] simplifies to #math.equation(block: false, alt: "the fraction y plus 2 over 5 .")[$frac(y + 2, 5) .$]

Now we will do this in reverse to split a single fraction into separate fractions. For example, #math.equation(block: false, alt: "the fraction y plus 2 over 5")[$frac(y + 2, 5)$] can be written #math.equation(block: false, alt: "the fraction y over 5 plus the fraction 2 over 5 .")[$frac(y, 5) + frac(2, 5) .$]

This is the “reverse” of fraction addition and it states that if #emph[a], #emph[b], and #emph[c] are numbers where #math.equation(block: false, alt: "c not equal to 0 ,")[$c ≠ 0 ,$] then #math.equation(block: false, alt: "the fraction a plus b over c equals the fraction a over c plus the fraction b over c .")[$frac(a + b, c) = frac(a, c) + frac(b, c) .$] We will use this to divide polynomials by monomials.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division of a Polynomial by a Monomial]

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
]

#examplebox("Example 3")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis 18 x cubed y minus 36 x y squared close parenthesis divided by open parenthesis −3 x y close parenthesis .")[$( 18 x^(3) y − 36 x y^(2) ) ÷ ( −3 x y ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 18 x cubed y minus 36 x y squared close parenthesis divided by open parenthesis −3 x y close parenthesis")[$( 18 x^(3) y − 36 x y^(2) ) ÷ ( −3 x y )$]]),
  [Rewrite as a fraction.], [#math.equation(block: false, alt: "the fraction 18 x cubed y minus 36 x y squared over −3 x y")[$frac(18 x^(3) y − 36 x y^(2), −3 x y)$]],
  [Divide each term by the divisor. Be careful with the signs!], [#math.equation(block: false, alt: "the fraction 18 x cubed y over −3 x y minus the fraction 36 x y squared over −3 x y")[$frac(18 x^(3) y, −3 x y) − frac(36 x y^(2), −3 x y)$]],
  [Simplify.], [#math.equation(block: false, alt: "−6 x squared plus 12 y")[$−6 x^(2) + 12 y$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 32 a squared b minus 16 a b squared close parenthesis divided by open parenthesis −8 a b close parenthesis .")[$( 32 a^(2) b − 16 a b^(2) ) ÷ ( −8 a b ) .$]

#solutionbox[
#math.equation(block: true, alt: "−4 a plus 2 b")[$−4 a + 2 b$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis −48 a to the power 8 b to the power 4 minus 36 a to the power 6 b to the power 5 close parenthesis divided by open parenthesis −6 a cubed b cubed close parenthesis .")[$( −48 a^(8) b^(4) − 36 a^(6) b^(5) ) ÷ ( −6 a^(3) b^(3) ) .$]

#solutionbox[
#math.equation(block: true, alt: "8 a to the power 5 b plus 6 a cubed b squared")[$8 a^(5) b + 6 a^(3) b^(2)$]
]
]


=== Divide Polynomials Using Long Division

Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

#figure(figph[This figure shows the long division of 875 divided by 25. 875 is labeled dividend and 25 is labeled divisor. The result of 35 is labeled quotient. The 3 in 35 is determined from the number of times we can divide 25 into 87. Multiplying 25 and 3 results in 75. 75 is subtracted from 87 to get 12. The 5 from 875 is dropped down to make 12 into 125. The 5 in 35 is determined from the number of times was can divide 25 into 125. Since 25 goes into 125 evenly there is no remainder. The result of subtracting 125 from 125 is 0 which is labeled remainder.], alt: "This figure shows the long division of 875 divided by 25. 875 is labeled dividend and 25 is labeled divisor. The result of 35 is labeled quotient. The 3 in 35 is determined from the number of times we can divide 25 into 87. Multiplying 25 and 3 results in 75. 75 is subtracted from 87 to get 12. The 5 from 875 is dropped down to make 12 into 125. The 5 in 35 is determined from the number of times was can divide 25 into 125. Since 25 goes into 125 evenly there is no remainder. The result of subtracting 125 from 125 is 0 which is labeled remainder.", caption: none)

We check division by multiplying the quotient by the divisor.

If we did the division correctly, the product should equal the dividend.

#math.equation(block: true, alt: "35 times 25; 875 ✓")[$35 · 25 \ 875 ✓$]Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.

#examplebox("Example 4")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis x squared plus 9 x plus 20 close parenthesis divided by open parenthesis x plus 5 close parenthesis .")[$( x^(2) + 9 x + 20 ) ÷ ( x + 5 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing polynomial division: (x^2 + 9x + 20) ÷ (x + 5).], alt: "A mathematical expression showing polynomial division: (x^2 + 9x + 20) ÷ (x + 5).", caption: none)]),
  [Write it as a long division problem. #linebreak() Be sure the dividend is in standard form.], [#figure(figph[A mathematical long division problem showing the polynomial x^2 + 9x + 20 being divided by the binomial x + 5.], alt: "A mathematical long division problem showing the polynomial x^2 + 9x + 20 being divided by the binomial x + 5.", caption: none)],
  [Divide #math.equation(block: false, alt: "x squared")[$x^(2)$] by #math.equation(block: false, alt: "x .")[$x .$] It may help to ask yourself, “What do I need #linebreak() to multiply #math.equation(block: false, alt: "x")[$x$] by to get #math.equation(block: false, alt: "x squared")[$x^(2)$]?”], [#figure(figph[Polynomial long division problem where x^2 + 9x + 20 is being divided by x + 5, with 'x' as the first term of the quotient.], alt: "Polynomial long division problem where x^2 + 9x + 20 is being divided by x + 5, with 'x' as the first term of the quotient.", caption: none)],
  [Put the answer, #math.equation(block: false, alt: "x ,")[$x ,$] in the quotient over the #math.equation(block: false, alt: "x")[$x$] term. #linebreak() Multiply #math.equation(block: false, alt: "x")[$x$] times #math.equation(block: false, alt: "x plus 5 .")[$x + 5 .$] Line up the like terms under the dividend.], [#figure(figph[A step in polynomial long division, showing (x+5) dividing into x^2 + 9x + 20. The term 'x' is placed in the quotient, and x(x+5) = x^2 + 5x is written below, ready for subtraction.], alt: "A step in polynomial long division, showing (x+5) dividing into x^2 + 9x + 20. The term 'x' is placed in the quotient, and x(x+5) = x^2 + 5x is written below, ready for subtraction.", caption: none)],
  [Subtract #math.equation(block: false, alt: "x squared plus 5 x")[$x^(2) + 5 x$] from #math.equation(block: false, alt: "x squared plus 9 x .")[$x^(2) + 9 x .$] #linebreak() You may find it easier to change the signs and then add. #linebreak() Then bring down the last term, 20.], [#figure(figph[Polynomial long division: dividing x^2 + 9x + 20 by x + 5. After placing 'x' in the quotient, -x^2 + (-5x) is subtracted from the dividend, resulting in 4x + 20, highlighted in red.], alt: "Polynomial long division: dividing x^2 + 9x + 20 by x + 5. After placing 'x' in the quotient, -x^2 + (-5x) is subtracted from the dividend, resulting in 4x + 20, highlighted in red.", caption: none)],
  [#linebreak() Divide #math.equation(block: false, alt: "4 x")[$4 x$] by #math.equation(block: false, alt: "x .")[$x .$] It may help to ask yourself, “What do I #linebreak() need to multiply #math.equation(block: false, alt: "x")[$x$] by to get #math.equation(block: false, alt: "4 x")[$4 x$]?” #linebreak() Put the answer, #math.equation(block: false, alt: "4")[$4$], in the quotient over the constant term.], [#figure(figph[Polynomial long division of (x^2 + 9x + 20) by (x + 5) is demonstrated, showing the initial steps to arrive at the quotient x + 4. The calculation subtracts (-x^2 + (-5x)) to get 4x + 20.], alt: "Polynomial long division of (x^2 + 9x + 20) by (x + 5) is demonstrated, showing the initial steps to arrive at the quotient x + 4. The calculation subtracts (-x^2 + (-5x)) to get 4x + 20.", caption: none)],
  [Multiply 4 times #math.equation(block: false, alt: "x plus 5 .")[$x + 5 .$]], [#figure(figph[Polynomial long division of x^2 + 9x + 20 by x + 5, yielding a quotient of x + 4. The steps illustrate the process of dividing polynomials, from the initial division to the final remainder.], alt: "Polynomial long division of x^2 + 9x + 20 by x + 5, yielding a quotient of x + 4. The steps illustrate the process of dividing polynomials, from the initial division to the final remainder.", caption: none)],
  [Subtract #math.equation(block: false, alt: "4 x plus 20")[$4 x + 20$] from #math.equation(block: false, alt: "4 x plus 20 .")[$4 x + 20 .$]], [#figure(figph[Polynomial long division problem for (x^2 + 9x + 20) divided by (x + 5), with the steps leading to a quotient of (x + 4) and a remainder of 0.], alt: "Polynomial long division problem for (x^2 + 9x + 20) divided by (x + 5), with the steps leading to a quotient of (x + 4) and a remainder of 0.", caption: none)],
  [Check: #linebreak() Multiply the quotient by the divisor. #math.equation(block: false, alt: "open parenthesis x plus 4 close parenthesis open parenthesis x plus 5 close parenthesis")[$#h(1em) ( x + 4 ) ( x + 5 )$] #linebreak() You should get the dividend. #math.equation(block: false, alt: "x squared plus 9 x plus 20 ✓")[$#h(4em) x^(2) + 9 x + 20 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis y squared plus 10 y plus 21 close parenthesis divided by open parenthesis y plus 3 close parenthesis .")[$( y^(2) + 10 y + 21 ) ÷ ( y + 3 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y plus 7")[$y + 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis m squared plus 9 m plus 20 close parenthesis divided by open parenthesis m plus 4 close parenthesis .")[$( m^(2) + 9 m + 20 ) ÷ ( m + 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "m plus 5")[$m + 5$]
]
]

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be #math.equation(block: false, alt: "x to the power 4 minus x squared plus 5 x minus 6 .")[$x^(4) − x^(2) + 5 x − 6 .$] It is missing an #math.equation(block: false, alt: "x cubed")[$x^(3)$] term. We will add in #math.equation(block: false, alt: "0 x cubed")[$0 x^(3)$] as a placeholder.

#examplebox("Example 5")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis x to the power 4 minus x squared plus 5 x minus 6 close parenthesis divided by open parenthesis x plus 2 close parenthesis .")[$( x^(4) − x^(2) + 5 x − 6 ) ÷ ( x + 2 ) .$]

#solutionbox[
Notice that there is no #math.equation(block: false, alt: "x cubed")[$x^(3)$] term in the dividend. We will add #math.equation(block: false, alt: "0 x cubed")[$0 x^(3)$] as a placeholder.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the division of a polynomial (x^4 - x^2 + 5x - 6) by a binomial (x + 2).], alt: "A mathematical expression showing the division of a polynomial (x^4 - x^2 + 5x - 6) by a binomial (x + 2).", caption: none)]),
  [Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.], [#figure(figph[A polynomial long division setup, showing x^4 + 0x^3 - x^2 + 5x - 6 being divided by x + 2, commonly used to find quotients and remainders of polynomial expressions.], alt: "A polynomial long division setup, showing x^4 + 0x^3 - x^2 + 5x - 6 being divided by x + 2, commonly used to find quotients and remainders of polynomial expressions.", caption: none)],
  [Divide #math.equation(block: false, alt: "x to the power 4")[$x^(4)$] by #math.equation(block: false, alt: "x .")[$x .$] #linebreak() Put the answer, #math.equation(block: false, alt: "x cubed ,")[$x^(3) ,$] in the quotient over the #math.equation(block: false, alt: "x cubed")[$x^(3)$] term. #linebreak() Multiply #math.equation(block: false, alt: "x cubed")[$x^(3)$] times #math.equation(block: false, alt: "x plus 2 .")[$x + 2 .$] Line up the like terms. #linebreak() Subtract and then bring down the next term.], [#figure(figph[A step in polynomial long division where x^4 + 0x^3 - x^2 + 5x - 6 is divided by x + 2, showing the subtraction of (x^4 + 2x^3) from x^4 + 0x^3 to get -2x^3 - x^2. A note suggests changing signs and adding.], alt: "A step in polynomial long division where x^4 + 0x^3 - x^2 + 5x - 6 is divided by x + 2, showing the subtraction of (x^4 + 2x^3) from x^4 + 0x^3 to get -2x^3 - x^2. A note suggests changing signs and adding.", caption: none)],
  [Divide #math.equation(block: false, alt: "−2 x cubed")[$−2 x^(3)$] by #math.equation(block: false, alt: "x .")[$x .$] #linebreak() Put the answer, #math.equation(block: false, alt: "−2 x squared ,")[$−2 x^(2) ,$] in the quotient over the #math.equation(block: false, alt: "x squared")[$x^(2)$] term. #linebreak() Multiply #math.equation(block: false, alt: "−2 x squared")[$−2 x^(2)$] times #math.equation(block: false, alt: "x plus 1 .")[$x + 1 .$] Line up the like terms #linebreak() Subtract and bring down the next term.], [#figure(figph[A visual guide to polynomial long division, showing the steps to divide x^4 - x^2 + 5x - 6 by x + 2. It highlights the subtraction of terms, noting that changing signs and adding can be helpful.], alt: "A visual guide to polynomial long division, showing the steps to divide x^4 - x^2 + 5x - 6 by x + 2. It highlights the subtraction of terms, noting that changing signs and adding can be helpful.", caption: none)],
  [Divide #math.equation(block: false, alt: "3 x squared")[$3 x^(2)$] by #math.equation(block: false, alt: "x .")[$x .$] #linebreak() Put the answer, #math.equation(block: false, alt: "3 x ,")[$3 x ,$] in the quotient over the #math.equation(block: false, alt: "x")[$x$] term. #linebreak() Multiply #math.equation(block: false, alt: "3 x")[$3 x$] times #math.equation(block: false, alt: "x plus 1 .")[$x + 1 .$] Line up the like terms. #linebreak() Subtract and bring down the next term.], [#figure(figph[This image illustrates polynomial long division, providing a step-by-step example. A helpful tip suggests changing the signs and adding instead of directly subtracting terms during the process.], alt: "This image illustrates polynomial long division, providing a step-by-step example. A helpful tip suggests changing the signs and adding instead of directly subtracting terms during the process.", caption: none)],
  [Divide #math.equation(block: false, alt: "− x")[$"−" x$] by #math.equation(block: false, alt: "x .")[$x .$] #linebreak() Put the answer, #math.equation(block: false, alt: "−1 ,")[$−1 ,$] in the quotient over the constant term. #linebreak() Multiply #math.equation(block: false, alt: "−1")[$−1$] times #math.equation(block: false, alt: "x plus 1 .")[$x + 1 .$] Line up the like terms. #linebreak() Change the signs, add. #linebreak()  #linebreak() Write the remainder as a fraction with the divisor as the denominator.], [#figure(figph[This image illustrates the process of polynomial long division, dividing x^4 - x^2 + 5x - 6 by x + 2, with a helpful reminder to change signs when subtracting.], alt: "This image illustrates the process of polynomial long division, dividing x^4 - x^2 + 5x - 6 by x + 2, with a helpful reminder to change signs when subtracting.", caption: none)],
  [To check, multiply #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x cubed minus 2 x squared plus 3 x minus 1 minus the fraction 4 over x plus 2 close parenthesis")[$( x + 2 ) ( x^(3) − 2 x^(2) + 3 x − 1 − frac(4, x + 2) )$]. #linebreak() The result should be #math.equation(block: false, alt: "x to the power 4 minus x squared plus 5 x minus 6 .")[$x^(4) − x^(2) + 5 x − 6 .$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis x to the power 4 minus 7 x squared plus 7 x plus 6 close parenthesis divided by open parenthesis x plus 3 close parenthesis .")[$( x^(4) − 7 x^(2) + 7 x + 6 ) ÷ ( x + 3 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x cubed minus 3 x squared plus 2 x plus 1 plus the fraction 3 over x plus 3")[$x^(3) − 3 x^(2) + 2 x + 1 + frac(3, x + 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis x to the power 4 minus 11 x squared minus 7 x minus 6 close parenthesis divided by open parenthesis x plus 3 close parenthesis .")[$( x^(4) − 11 x^(2) − 7 x − 6 ) ÷ ( x + 3 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x cubed minus 3 x squared minus 2 x minus 1 minus the fraction 3 over x plus 3")[$x^(3) − 3 x^(2) − 2 x − 1 − frac(3, x + 3)$]
]
]

In the next example, we will divide by #math.equation(block: false, alt: "2 a plus 3 .")[$2 a + 3 .$] As we divide, we will have to consider the constants as well as the variables.

#examplebox("Example 6")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis 8 a cubed plus 27 close parenthesis divided by open parenthesis 2 a plus 3 close parenthesis .")[$( 8 a^(3) + 27 ) ÷ ( 2 a + 3 ) .$]

#solutionbox[
This time we will show the division all in one step. We need to add two placeholders in order to divide.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the division of a sum of cubes, (8a^3 + 27), by a binomial, (2a + 3), written horizontally with a division symbol.], alt: "A mathematical expression showing the division of a sum of cubes, (8a^3 + 27), by a binomial, (2a + 3), written horizontally with a division symbol.", caption: none)]),
  [], [#figure(figph[A step-by-step example of polynomial long division showing (8a^3 + 27) divided by (2a + 3), resulting in 4a^2 - 6a + 9 with a remainder of 0. The intermediate multiplication steps are also indicated.], alt: "A step-by-step example of polynomial long division showing (8a^3 + 27) divided by (2a + 3), resulting in 4a^2 - 6a + 9 with a remainder of 0. The intermediate multiplication steps are also indicated.", caption: none)],
))

To check, multiply #math.equation(block: false, alt: "open parenthesis 2 a plus 3 close parenthesis open parenthesis 4 a squared minus 6 a plus 9 close parenthesis .")[$( 2 a + 3 ) ( 4 a^(2) − 6 a + 9 ) .$]

The result should be #math.equation(block: false, alt: "8 a cubed plus 27 .")[$8 a^(3) + 27 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis x cubed minus 64 close parenthesis divided by open parenthesis x minus 4 close parenthesis .")[$( x^(3) − 64 ) ÷ ( x − 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x squared plus 4 x plus 16")[$x^(2) + 4 x + 16$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 125 x cubed minus 8 close parenthesis divided by open parenthesis 5 x minus 2 close parenthesis .")[$( 125 x^(3) − 8 ) ÷ ( 5 x − 2 ) .$]

#solutionbox[
#math.equation(block: true, alt: "25 x squared plus 10 x plus 4")[$25 x^(2) + 10 x + 4$]
]
]


=== Divide Polynomials using Synthetic Division

As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in  and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.

#figure(figph[The figure shows the long division of 1 x squared plus 9 x plus 20 divided by x plus 5 right next to the same problem done with synthetic division. In the long division problem, the coefficients of the dividend are 1 and 9 and 20 and the zero of the divisor is negative 5. In the synthetic division problem, we just write the numbers negative 5 1 9 20 with a line separating the negative 5. In the long division problem, the subtracted terms are 5 x and 20. In the synthetic division problem the second line is the numbers negative 5 and negative 20. The remainder of the problem is 0 and the quotient is x plus 4. The synthetic division puts these coefficients as the last line 1 4 0.], alt: "The figure shows the long division of 1 x squared plus 9 x plus 20 divided by x plus 5 right next to the same problem done with synthetic division. In the long division problem, the coefficients of the dividend are 1 and 9 and 20 and the zero of the divisor is negative 5. In the synthetic division problem, we just write the numbers negative 5 1 9 20 with a line separating the negative 5. In the long division problem, the subtracted terms are 5 x and 20. In the synthetic division problem the second line is the numbers negative 5 and negative 20. The remainder of the problem is 0 and the quotient is x plus 4. The synthetic division puts these coefficients as the last line 1 4 0.", caption: none)

Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "x squared")[$x^(2)$] are removed. as well as the #math.equation(block: false, alt: "− x squared")[$"−" x^(2)$] and #math.equation(block: false, alt: "−4 x")[$−4 x$] as they are opposite the term above.

The first row of the synthetic division is the coefficients of the dividend. The #math.equation(block: false, alt: "−5")[$−5$] is the opposite of the 5 in the divisor.

The second row of the synthetic division are the numbers shown in red in the division problem.

The third row of the synthetic division are the numbers shown in blue in the division problem.

Notice the quotient and remainder are shown in the third row.

#math.equation(block: true, alt: "Synthetic division only works when the divisor is of the form x minus c .")[$"Synthetic division only works when the divisor is of the form" #h(0.2em) x − c .$]The following example will explain the process.

#examplebox("Example 7")[][
Use synthetic division to find the quotient and remainder when #math.equation(block: false, alt: "2 x cubed plus 3 x squared plus x plus 8")[$2 x^(3) + 3 x^(2) + x + 8$] is divided by #math.equation(block: false, alt: "x plus 2 .")[$x + 2 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Write the dividend with decreasing powers of #math.equation(block: false, alt: "x .")[$x .$]], [#figure(figph[The polynomial expression 2x^3 + 3x^2 + x + 8.], alt: "The polynomial expression 2x^3 + 3x^2 + x + 8.", caption: none)]),
  [Write the coefficients of the terms as the first #linebreak() row of the synthetic division.], [#figure(figph[A sequence of numbers 2, 3, 1, 8 is displayed on a white background, partially enclosed by a dark gray bracket on the left.], alt: "A sequence of numbers 2, 3, 1, 8 is displayed on a white background, partially enclosed by a dark gray bracket on the left.", caption: none)],
  [Write the divisor as #math.equation(block: false, alt: "x minus c")[$x − c$] and place #emph[c] #linebreak() in the synthetic division in the divisor box.], [#figure(figph[A synthetic division problem showing -2 as the divisor and 2, 3, 1, 8 as the coefficients of a polynomial.], alt: "A synthetic division problem showing -2 as the divisor and 2, 3, 1, 8 as the coefficients of a polynomial.", caption: none)],
  [Bring down the first coefficient to the third row.], [#figure(figph[Illustrates the initial step of synthetic division: the first coefficient (2) of the polynomial is brought down below the line, with -2 as the divisor and 3, 1, 8 as subsequent coefficients.], alt: "Illustrates the initial step of synthetic division: the first coefficient (2) of the polynomial is brought down below the line, with -2 as the divisor and 3, 1, 8 as subsequent coefficients.", caption: none)],
  [Multiply that coefficient by the divisor and place the #linebreak() result in the second row under the second coefficient.], [#figure(figph[An image showing the initial steps of synthetic division. The number -2 is being divided into a polynomial with coefficients 2, 3, 1, and 8. The leading coefficient, 2, is brought down, then multiplied by -2 to get -4.], alt: "An image showing the initial steps of synthetic division. The number -2 is being divided into a polynomial with coefficients 2, 3, 1, and 8. The leading coefficient, 2, is brought down, then multiplied by -2 to get -4.", caption: none)],
  [Add the second column, putting the result in the third row.], [#figure(figph[A mathematical division problem is depicted, likely synthetic division, showing the numbers -2, 2, 3, 1, 8. Below the 3, -4 is written, and below a horizontal line, -1 appears with a light blue arrow pointing down from 3 to -4 and then to -1, indicating subtraction. The number 2 is also shown below the first horizontal line.], alt: "A mathematical division problem is depicted, likely synthetic division, showing the numbers -2, 2, 3, 1, 8. Below the 3, -4 is written, and below a horizontal line, -1 appears with a light blue arrow pointing down from 3 to -4 and then to -1, indicating subtraction. The number 2 is also shown below the first horizontal line.", caption: none)],
  [Multiply that result by the divisor and place the #linebreak() result in the second row under the third coefficient.], [#figure(figph[An image illustrating the initial steps of synthetic division, showing coefficients (2, 3, 1, 8) divided by -2, with multiplication and addition leading to the first few quotient terms.], alt: "An image illustrating the initial steps of synthetic division, showing coefficients (2, 3, 1, 8) divided by -2, with multiplication and addition leading to the first few quotient terms.", caption: none)],
  [Add the third column, putting the result in the third row.], [#figure(figph[A step in synthetic division showing the coefficients 2, 3, 1, and 8 being divided by -2. The process shows bringing down 2, multiplying by -2 to get -4, then adding 3 and -4 to get -1. Next, -1 is multiplied by -2 to get 2, and 1 plus 2 yields 3.], alt: "A step in synthetic division showing the coefficients 2, 3, 1, and 8 being divided by -2. The process shows bringing down 2, multiplying by -2 to get -4, then adding 3 and -4 to get -1. Next, -1 is multiplied by -2 to get 2, and 1 plus 2 yields 3.", caption: none)],
  [Multiply that result by the divisor and place the #linebreak() result in the third row under the third coefficient.], [#figure(figph[A mathematical example demonstrating the process of synthetic division, with a divisor of -2 and polynomial coefficients 2, 3, 1, and 8. Blue arrows indicate the steps of multiplication and addition.], alt: "A mathematical example demonstrating the process of synthetic division, with a divisor of -2 and polynomial coefficients 2, 3, 1, and 8. Blue arrows indicate the steps of multiplication and addition.", caption: none)],
  [Add the final column, putting the result in the third row.], [#figure(figph[An example of synthetic division is shown, with -2 as the divisor and 2, 3, 1, 8 as the dividend coefficients. The result is a quotient with coefficients 2, -1, 3 and a remainder of 2.], alt: "An example of synthetic division is shown, with -2 as the divisor and 2, 3, 1, 8 as the dividend coefficients. The result is a quotient with coefficients 2, -1, 3 and a remainder of 2.", caption: none)],
  [The quotient is #math.equation(block: false, alt: "2 x squared minus 1 x plus 3")[$2 x^(2) − 1 x + 3$] and the remainder is 2.], [],
))

The division is complete. The numbers in the third row give us the result. The #math.equation(block: false, alt: "2 −1 3")[$2 #h(1em) −1 #h(1em) 3$] are the coefficients of the quotient. The quotient is #math.equation(block: false, alt: "2 x squared minus 1 x plus 3 .")[$2 x^(2) − 1 x + 3 .$] The 2 in the box in the third row is the remainder.

Check:

#math.equation(block: true, alt: "(quotient)(divisor) plus remainder, equals, dividend; open parenthesis 2 x squared minus 1 x plus 3 close parenthesis open parenthesis x plus 2 close parenthesis plus 2, equals ?, 2 x cubed plus 3 x squared plus x plus 8; 2 x cubed minus x squared plus 3 x plus 4 x squared minus 2 x plus 6 plus 2, equals ?, 2 x cubed plus 3 x squared plus x plus 8; 2 x cubed plus 3 x squared plus x plus 8, equals, 2 x cubed plus 3 x squared plus x plus 8 ✓")[$#h(2em) "(quotient)(divisor)" + "remainder" & = & "dividend" \ ( 2 x^(2) − 1 x + 3 ) ( x + 2 ) + 2 & limits(=)^(?) & 2 x^(3) + 3 x^(2) + x + 8 \ 2 x^(3) − x^(2) + 3 x + 4 x^(2) − 2 x + 6 + 2 & limits(=)^(?) & 2 x^(3) + 3 x^(2) + x + 8 \ 2 x^(3) + 3 x^(2) + x + 8 & = & 2 x^(3) + 3 x^(2) + x + 8 ✓$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use synthetic division to find the quotient and remainder when #math.equation(block: false, alt: "3 x cubed plus 10 x squared plus 6 x minus 2")[$3 x^(3) + 10 x^(2) + 6 x − 2$] is divided by #math.equation(block: false, alt: "x plus 2 .")[$x + 2 .$]

#solutionbox[
#math.equation(block: true, alt: "3 x squared plus 4 x minus 2 ; 2")[$3 x^(2) + 4 x − 2 ";" #h(0.2em) 2$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use synthetic division to find the quotient and remainder when #math.equation(block: false, alt: "4 x cubed plus 5 x squared minus 5 x plus 3")[$4 x^(3) + 5 x^(2) − 5 x + 3$] is divided by #math.equation(block: false, alt: "x plus 2 .")[$x + 2 .$]

#solutionbox[
#math.equation(block: true, alt: "4 x squared minus 3 x plus 1 ; 1")[$4 x^(2) − 3 x + 1 ";" #h(0.2em) 1$]
]
]

In the next example, we will do all the steps together.

#examplebox("Example 8")[][
Use synthetic division to find the quotient and remainder when #math.equation(block: false, alt: "x to the power 4 minus 16 x squared plus 3 x plus 12")[$x^(4) − 16 x^(2) + 3 x + 12$] is divided by #math.equation(block: false, alt: "x plus 4 .")[$x + 4 .$]

#solutionbox[
The polynomial #math.equation(block: false, alt: "x to the power 4 minus 16 x squared plus 3 x plus 12")[$x^(4) − 16 x^(2) + 3 x + 12$] has its term in order with descending degree but we notice there is no #math.equation(block: false, alt: "x cubed")[$x^(3)$] term. We will add a 0 as a placeholder for the #math.equation(block: false, alt: "x cubed")[$x^(3)$] term. In #math.equation(block: false, alt: "x minus c")[$x − c$] form, the divisor is #math.equation(block: false, alt: "x minus open parenthesis −4 close parenthesis .")[$x − ( −4 ) .$]

#figure(figph[The figure shows the results of using synthetic division with the example of the polynomial x to the fourth power minus 16 x squared plus 3 x plus 12 divided by x plus 4. The divisor number if negative 4. The first row is 1 0 negative 16 3 12. The first column is 1 blank 1. The second column is negative 16 16 0. The third column is 3 0 3. The fourth column is 12 negative 12 0.], alt: "The figure shows the results of using synthetic division with the example of the polynomial x to the fourth power minus 16 x squared plus 3 x plus 12 divided by x plus 4. The divisor number if negative 4. The first row is 1 0 negative 16 3 12. The first column is 1 blank 1. The second column is negative 16 16 0. The third column is 3 0 3. The fourth column is 12 negative 12 0.", caption: none)

We divided a 4#super[th] degree polynomial by a 1#super[st] degree polynomial so the quotient will be a 3#super[rd] degree polynomial.

Reading from the third row, the quotient has the coefficients #math.equation(block: false, alt: "1 −4 0 3 ,")[$1 #h(1em) −4 #h(1em) 0 #h(1em) 3 ,$] which is #math.equation(block: false, alt: "x cubed minus 4 x squared plus 3 .")[$x^(3) − 4 x^(2) + 3 .$] The remainder #linebreak() is 0.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use synthetic division to find the quotient and remainder when #math.equation(block: false, alt: "x to the power 4 minus 16 x squared plus 5 x plus 20")[$x^(4) − 16 x^(2) + 5 x + 20$] is divided by #math.equation(block: false, alt: "x plus 4 .")[$x + 4 .$]

#solutionbox[
#math.equation(block: true, alt: "x cubed minus 4 x squared plus 5 ; 0")[$x^(3) − 4 x^(2) + 5 ";" #h(0.2em) 0$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use synthetic division to find the quotient and remainder when #math.equation(block: false, alt: "x to the power 4 minus 9 x squared plus 2 x plus 6")[$x^(4) − 9 x^(2) + 2 x + 6$] is divided by #math.equation(block: false, alt: "x plus 3 .")[$x + 3 .$]

#solutionbox[
#math.equation(block: true, alt: "x cubed minus 3 x squared plus 2 ; 0")[$x^(3) − 3 x^(2) + 2 ";" #h(0.2em) 0$]
]
]


=== Divide Polynomial Functions

Just as polynomials can be divided, polynomial functions can also be divided.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division of Polynomial Functions]

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis ,")[$g ( x ) ,$] where #math.equation(block: false, alt: "g open parenthesis x close parenthesis not equal to 0 ,")[$g ( x ) ≠ 0 ,$]

#math.equation(block: true, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals the fraction f open parenthesis x close parenthesis over g open parenthesis x close parenthesis")[$( frac(f, g) ) ( x ) = frac(f ( x ), g ( x ))$]
]

#examplebox("Example 9")[][
For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 5 x minus 14")[$f ( x ) = x^(2) − 5 x − 14$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x plus 2 ,")[$g ( x ) = x + 2 ,$] find: ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −4 close parenthesis .")[$( frac(f, g) ) ( −4 ) .$]

#solutionbox[
ⓐ  #linebreak() 
#figure(figph[Equation shows f over g of x equals f of x divided by g of x. This is translated into a division problem showing x squared minus 5x minus 14 divided by x plus 2. The quotient is x minus 7.], alt: "Equation shows f over g of x equals f of x divided by g of x. This is translated into a division problem showing x squared minus 5x minus 14 divided by x plus 2. The quotient is x minus 7.", caption: none)

 #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Substitute for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis .")[$g ( x ) .$]], [#math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals the fraction x squared minus 5 x minus 14 over x plus 2")[$#h(4.6em) ( frac(f, g) ) ( x ) = frac(x^(2) − 5 x − 14, x + 2)$]]),
  [Divide the polynomials.], [#math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals x minus 7")[$#h(4.6em) ( frac(f, g) ) ( x ) = x − 7$]],
))

#linebreak() 
ⓑ In part ⓐ we found #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] and now are asked to find #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −4 close parenthesis .")[$( frac(f, g) ) ( −4 ) .$] #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals x minus 7")[$#h(1em) ( frac(f, g) ) ( x ) = x − 7$]]),
  [To find #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −4 close parenthesis ,")[$( frac(f, g) ) ( −4 ) ,$] substitute #math.equation(block: false, alt: "x equals −4 .")[$x = −4 .$]], [#math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −4 close parenthesis equals −4 minus 7")[$( frac(f, g) ) ( −4 ) = −4 − 7$]],
  [], [#math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −4 close parenthesis equals −11")[$( frac(f, g) ) ( −4 ) = −11$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 5 x minus 24")[$f ( x ) = x^(2) − 5 x − 24$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x plus 3 ,")[$g ( x ) = x + 3 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −3 close parenthesis .")[$( frac(f, g) ) ( −3 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals x minus 8")[$( frac(f, g) ) ( x ) = x − 8$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −3 close parenthesis equals −11")[$( frac(f, g) ) ( −3 ) = −11$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 5 x minus 36")[$f ( x ) = x^(2) − 5 x − 36$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x plus 4 ,")[$g ( x ) = x + 4 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −5 close parenthesis .")[$( frac(f, g) ) ( −5 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals x minus 9")[$( frac(f, g) ) ( x ) = x − 9$] #linebreak() ⓑ undefined
]
]


=== Use the Remainder and Factor Theorem

Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as #math.equation(block: false, alt: "x minus c ,")[$x − c ,$] the value of the function at #math.equation(block: false, alt: "c , f open parenthesis c close parenthesis ,")[$c , f ( c ) ,$] is the same as the remainder from the division problem.

#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Dividend], [Divisor #math.equation(block: false, alt: "x minus c")[$x − c$]], [Remainder], [Function], [#math.equation(block: false, alt: "f open parenthesis c close parenthesis")[$f ( c )$]]),
  [#math.equation(block: false, alt: "x to the power 4 minus x squared plus 5 x minus 6")[$x^(4) − x^(2) + 5 x − 6$]], [#math.equation(block: false, alt: "x minus open parenthesis −2 close parenthesis")[$x − ( −2 )$]], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 4 minus x squared plus 5 x minus 6")[$f ( x ) = x^(4) − x^(2) + 5 x − 6$]], [#math.equation(block: false, alt: "−4")[$−4$]],
  [#math.equation(block: false, alt: "3 x cubed minus 2 x squared minus 10 x plus 8")[$3 x^(3) − 2 x^(2) − 10 x + 8$]], [#math.equation(block: false, alt: "x minus 2")[$x − 2$]], [4], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 x cubed minus 2 x squared minus 10 x plus 8")[$f ( x ) = 3 x^(3) − 2 x^(2) − 10 x + 8$]], [4],
  [#math.equation(block: false, alt: "x to the power 4 minus 16 x squared plus 3 x plus 15")[$x^(4) − 16 x^(2) + 3 x + 15$]], [#math.equation(block: false, alt: "x minus open parenthesis −4 close parenthesis")[$x − ( −4 )$]], [3], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 4 minus 16 x squared plus 3 x plus 15")[$f ( x ) = x^(4) − 16 x^(2) + 3 x + 15$]], [3],
))

To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend #math.equation(block: false, alt: "f open parenthesis x close parenthesis ,")[$f ( x ) ,$] we multiply the quotient, #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$] times the divisor, #math.equation(block: false, alt: "x minus c ,")[$x − c ,$] and add the remainder, #emph[r].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation illustrating the Remainder Theorem: f(x) = q(x)(x - c) + r, where f(x) is a polynomial, q(x) is the quotient, (x-c) is the divisor, and r is the remainder.], alt: "A mathematical equation illustrating the Remainder Theorem: f(x) = q(x)(x - c) + r, where f(x) is a polynomial, q(x) is the quotient, (x-c) is the divisor, and r is the remainder.", caption: none)]),
  [If we evaluate this at #math.equation(block: false, alt: "c ,")[$c ,$] we get:], [#figure(figph[A mathematical equation is displayed: f(c) = q(c)(c - c) + r. The variable 'c' is highlighted in red within f(c), q(c), and (c - c).], alt: "A mathematical equation is displayed: f(c) = q(c)(c - c) + r. The variable 'c' is highlighted in red within f(c), q(c), and (c - c).", caption: none)],
  [], [#figure(figph[A mathematical equation showing f(c) = q(c)(0) + r, where the term q(c)(0) simplifies to zero, resulting in f(c) = r.], alt: "A mathematical equation showing f(c) = q(c)(0) + r, where the term q(c)(0) simplifies to zero, resulting in f(c) = r.", caption: none)],
  [], [#figure(figph[The mathematical equation f(c) = r is displayed on a white background.], alt: "The mathematical equation f(c) = r is displayed on a white background.", caption: none)],
))

This leads us to the Remainder Theorem.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Remainder Theorem]

If the polynomial function #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is divided by #math.equation(block: false, alt: "x minus c ,")[$x − c ,$] then the remainder is #math.equation(block: false, alt: "f open parenthesis c close parenthesis .")[$f ( c ) .$]
]

#examplebox("Example 10")[][
Use the Remainder Theorem to find the remainder when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus 3 x plus 19")[$f ( x ) = x^(3) + 3 x + 19$] is divided by #math.equation(block: false, alt: "x plus 2 .")[$x + 2 .$]

#solutionbox[
To use the Remainder Theorem, we must use the divisor in the #math.equation(block: false, alt: "x minus c")[$x − c$] form. We can write the divisor #math.equation(block: false, alt: "x plus 2")[$x + 2$] as #math.equation(block: false, alt: "x minus open parenthesis −2 close parenthesis .")[$x − ( −2 ) .$] So, our #math.equation(block: false, alt: "c")[$c$] is #math.equation(block: false, alt: "−2 .")[$−2 .$]

To find the remainder, we evaluate #math.equation(block: false, alt: "f open parenthesis c close parenthesis")[$f ( c )$] which is #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis .")[$f ( −2 ) .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the mathematical function f(x) = x³ + 3x + 19.], alt: "The image shows the mathematical function f(x) = x³ + 3x + 19.", caption: none)]),
  [To evaluate #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis ,")[$f ( −2 ) ,$] substitute #math.equation(block: false, alt: "x equals −2 .")[$x = −2 .$]], [#figure(figph[A mathematical equation is displayed: f(-2) = (-2)^3 + 3(-2) + 19, showing the substitution of -2 into a function.], alt: "A mathematical equation is displayed: f(-2) = (-2)^3 + 3(-2) + 19, showing the substitution of -2 into a function.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed against a white background, reading 'f(-2) = -8 - 6 + 19'.], alt: "A mathematical equation is displayed against a white background, reading 'f(-2) = -8 - 6 + 19'.", caption: none)],
  [], [#figure(figph[A mathematical equation displays 'f(-2) = 5' in a clear, bold font on a white background, representing the function f evaluated at -2 equals 5.], alt: "A mathematical equation displays 'f(-2) = 5' in a clear, bold font on a white background, representing the function f evaluated at -2 equals 5.", caption: none)],
  [], [The remainder is 5 when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus 3 x plus 19")[$f ( x ) = x^(3) + 3 x + 19$] is divided by #math.equation(block: false, alt: "x plus 2 .")[$x + 2 .$]],
  [Check: #linebreak() Use synthetic division to check.], [],
  [#figure(figph[A synthetic division problem with a divisor of -2. The dividend's coefficients are 1, 0, 3, 19. The result shows quotient coefficients 1, -2, 7, and a remainder of 5.], alt: "A synthetic division problem with a divisor of -2. The dividend's coefficients are 1, 0, 3, 19. The result shows quotient coefficients 1, -2, 7, and a remainder of 5.", caption: none)], [],
  [The remainder is 5.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Remainder Theorem to find the remainder when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus 4 x plus 15")[$f ( x ) = x^(3) + 4 x + 15$] is divided by #math.equation(block: false, alt: "x plus 2 .")[$x + 2 .$]

#solutionbox[
#math.equation(block: true, alt: "−1")[$−1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Remainder Theorem to find the remainder when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 7 x plus 12")[$f ( x ) = x^(3) − 7 x + 12$] is divided by #math.equation(block: false, alt: "x plus 3 .")[$x + 3 .$]

#solutionbox[
#math.equation(block: true, alt: "6")[$6$]
]
]

When we divided #math.equation(block: false, alt: "8 a cubed plus 27")[$8 a^(3) + 27$] by #math.equation(block: false, alt: "2 a plus 3")[$2 a + 3$] in  the result was #math.equation(block: false, alt: "4 a squared minus 6 a plus 9 .")[$4 a^(2) − 6 a + 9 .$] To check our work, we multiply #math.equation(block: false, alt: "4 a squared minus 6 a plus 9")[$4 a^(2) − 6 a + 9$] by #math.equation(block: false, alt: "2 a plus 3")[$2 a + 3$] to get #math.equation(block: false, alt: "8 a cubed plus 27")[$8 a^(3) + 27$].

#math.equation(block: true, alt: "open parenthesis 4 a squared minus 6 a plus 9 close parenthesis open parenthesis 2 a plus 3 close parenthesis equals 8 a cubed plus 27")[$( 4 a^(2) − 6 a + 9 ) ( 2 a + 3 ) = 8 a^(3) + 27$]Written this way, we can see that #math.equation(block: false, alt: "4 a squared minus 6 a plus 9")[$4 a^(2) − 6 a + 9$] and #math.equation(block: false, alt: "2 a plus 3")[$2 a + 3$] are factors of #math.equation(block: false, alt: "8 a cubed plus 27 .")[$8 a^(3) + 27 .$] When we did the division, the remainder was zero.

Whenever a divisor, #math.equation(block: false, alt: "x minus c ,")[$x − c ,$] divides a polynomial function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis ,")[$f ( x ) ,$] and resulting in a remainder of zero, we say #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis .")[$f ( x ) .$]

The reverse is also true. If #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] then #math.equation(block: false, alt: "x minus c")[$x − c$] will divide the polynomial function resulting in a remainder of zero.

We will state this in the Factor Theorem.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor Theorem]

For any polynomial function #math.equation(block: false, alt: "f open parenthesis x close parenthesis ,")[$f ( x ) ,$]

- if #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis ,")[$f ( x ) ,$] then #math.equation(block: false, alt: "f open parenthesis c close parenthesis equals 0")[$f ( c ) = 0$]
- if #math.equation(block: false, alt: "f open parenthesis c close parenthesis equals 0 ,")[$f ( c ) = 0 ,$] then #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]
]

#examplebox("Example 11")[][
Use the Remainder Theorem to determine if #math.equation(block: false, alt: "x minus 4")[$x − 4$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 64 .")[$f ( x ) = x^(3) − 64 .$]

#solutionbox[
The Factor Theorem tells us that #math.equation(block: false, alt: "x minus 4")[$x − 4$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 64")[$f ( x ) = x^(3) − 64$] if #math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals 0 .")[$f ( 4 ) = 0 .$] #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 64")[$f ( x ) = x^(3) − 64$]]),
  [To evaluate #math.equation(block: false, alt: "f open parenthesis 4 close parenthesis")[$f ( 4 )$] substitute #math.equation(block: false, alt: "x equals 4 .")[$x = 4 .$]], [#math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals 4 cubed minus 64")[$f ( 4 ) = 4^(3) − 64$]],
  [Simplify.], [#math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals 64 minus 64")[$f ( 4 ) = 64 − 64$]],
  [Subtract.], [#math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals 0")[$f ( 4 ) = 0$]],
))

Since #math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals 0 ,")[$f ( 4 ) = 0 ,$] #math.equation(block: false, alt: "x minus 4")[$x − 4$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 64 .")[$f ( x ) = x^(3) − 64 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Factor Theorem to determine if #math.equation(block: false, alt: "x minus 5")[$x − 5$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 125 .")[$f ( x ) = x^(3) − 125 .$]

#solutionbox[
yes
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Factor Theorem to determine if #math.equation(block: false, alt: "x minus 6")[$x − 6$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 216 .")[$f ( x ) = x^(3) − 216 .$]

#solutionbox[
yes
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with dividing polynomials.

- #link("https://openstax.org/l/37Polybybinom")[Dividing a Polynomial by a Binomial]
- #link("https://openstax.org/l/37SynDivision")[Synthetic Division & Remainder Theorem]
]


=== Key Concepts

- #strong[Division of a Polynomial by a Monomial]
- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
- #strong[Division of Polynomial Functions]
- For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis ,")[$g ( x ) ,$] where #math.equation(block: false, alt: "g open parenthesis x close parenthesis not equal to 0 ,")[$g ( x ) ≠ 0 ,$] #linebreak() 
#math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals the fraction f open parenthesis x close parenthesis over g open parenthesis x close parenthesis")[$( frac(f, g) ) ( x ) = frac(f ( x ), g ( x ))$]
- #strong[Remainder Theorem]
- If the polynomial function #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is divided by #math.equation(block: false, alt: "x minus c ,")[$x − c ,$] then the remainder is #math.equation(block: false, alt: "f open parenthesis c close parenthesis .")[$f ( c ) .$]
- #strong[Factor Theorem:] For any polynomial function #math.equation(block: false, alt: "f open parenthesis x close parenthesis ,")[$f ( x ) ,$]
- if #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis ,")[$f ( x ) ,$] then #math.equation(block: false, alt: "f open parenthesis c close parenthesis equals 0")[$f ( c ) = 0$]
- if #math.equation(block: false, alt: "f open parenthesis c close parenthesis equals 0 ,")[$f ( c ) = 0 ,$] then #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]


=== Section Exercises


==== Practice Makes Perfect

#strong[Divide Monomials]

In the following exercises, divide the monomials.

#math.equation(block: true, alt: "15 r to the power 4 s to the power 9 divided by open parenthesis 15 r to the power 4 s to the power 9 close parenthesis")[$15 r^(4) s^(9) ÷ ( 15 r^(4) s^(9) )$]

#math.equation(block: true, alt: "20 m to the power 8 n to the power 4 divided by open parenthesis 30 m to the power 5 n to the power 9 close parenthesis")[$20 m^(8) n^(4) ÷ ( 30 m^(5) n^(9) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 m cubed over 3 n to the power 5")[$frac(2 m^(3), 3 n^(5))$]
]

#math.equation(block: true, alt: "the fraction 18 a to the power 4 b to the power 8 over −27 a to the power 9 b to the power 5")[$frac(18 a^(4) b^(8), −27 a^(9) b^(5))$]

#math.equation(block: true, alt: "the fraction 45 x to the power 5 y to the power 9 over −60 x to the power 8 y to the power 6")[$frac(45 x^(5) y^(9), −60 x^(8) y^(6))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction −3 y cubed over 4 x cubed")[$frac(−3 y^(3), 4 x^(3))$]
]

#math.equation(block: true, alt: "the fraction open parenthesis 10 m to the power 5 n to the power 4 close parenthesis open parenthesis 5 m cubed n to the power 6 close parenthesis over 25 m to the power 7 n to the power 5")[$frac(( 10 m^(5) n^(4) ) ( 5 m^(3) n^(6) ), 25 m^(7) n^(5))$]

#math.equation(block: true, alt: "the fraction open parenthesis −18 p to the power 4 q to the power 7 close parenthesis open parenthesis −6 p cubed q to the power 8 close parenthesis over −36 p to the power 12 q to the power 10")[$frac(( −18 p^(4) q^(7) ) ( −6 p^(3) q^(8) ), −36 p^(12) q^(10))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction −3 q to the power 5 over p to the power 5")[$frac(−3 q^(5), p^(5))$]
]

#math.equation(block: true, alt: "the fraction open parenthesis 6 a to the power 4 b cubed close parenthesis open parenthesis 4 a b to the power 5 close parenthesis over open parenthesis 12 a squared b close parenthesis open parenthesis a cubed b close parenthesis")[$frac(( 6 a^(4) b^(3) ) ( 4 a b^(5) ), ( 12 a^(2) b ) ( a^(3) b ))$]

#math.equation(block: true, alt: "the fraction open parenthesis 4 u squared v to the power 5 close parenthesis open parenthesis 15 u cubed v close parenthesis over open parenthesis 12 u cubed v close parenthesis open parenthesis u to the power 4 v close parenthesis")[$frac(( 4 u^(2) v^(5) ) ( 15 u^(3) v ), ( 12 u^(3) v ) ( u^(4) v ))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 v to the power 4 over u squared")[$frac(5 v^(4), u^(2))$]
]

#strong[Divide a Polynomial by a Monomial]

In the following exercises, divide each polynomial by the monomial.

#math.equation(block: true, alt: "open parenthesis 9 n to the power 4 plus 6 n cubed close parenthesis divided by 3 n")[$( 9 n^(4) + 6 n^(3) ) ÷ 3 n$]

#math.equation(block: true, alt: "open parenthesis 8 x cubed plus 6 x squared close parenthesis divided by 2 x")[$( 8 x^(3) + 6 x^(2) ) ÷ 2 x$]

#solutionbox[
#math.equation(block: true, alt: "4 x squared plus 3 x")[$4 x^(2) + 3 x$]
]

#math.equation(block: true, alt: "open parenthesis 63 m to the power 4 minus 42 m cubed close parenthesis divided by open parenthesis −7 m squared close parenthesis")[$( 63 m^(4) − 42 m^(3) ) ÷ ( −7 m^(2) )$]

#math.equation(block: true, alt: "open parenthesis 48 y to the power 4 minus 24 y cubed close parenthesis divided by open parenthesis −8 y squared close parenthesis")[$( 48 y^(4) − 24 y^(3) ) ÷ ( −8 y^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "−6 y squared plus 3 y")[$−6 y^(2) + 3 y$]
]

#math.equation(block: true, alt: "the fraction 66 x cubed y squared minus 110 x squared y cubed minus 44 x to the power 4 y cubed over 11 x squared y squared")[$frac(66 x^(3) y^(2) − 110 x^(2) y^(3) − 44 x^(4) y^(3), 11 x^(2) y^(2))$]

#math.equation(block: true, alt: "the fraction 72 r to the power 5 s squared plus 132 r to the power 4 s cubed minus 96 r cubed s to the power 5 over 12 r squared s squared")[$frac(72 r^(5) s^(2) + 132 r^(4) s^(3) − 96 r^(3) s^(5), 12 r^(2) s^(2))$]

#solutionbox[
#math.equation(block: true, alt: "6 r cubed plus 11 r squared s minus 8 r s cubed")[$6 r^(3) + 11 r^(2) s − 8 r s^(3)$]
]

#math.equation(block: true, alt: "the fraction 10 x squared plus 5 x minus 4 over −5 x")[$frac(10 x^(2) + 5 x − 4, −5 x)$]

#math.equation(block: true, alt: "the fraction 20 y squared plus 12 y minus 1 over −4 y")[$frac(20 y^(2) + 12 y − 1, −4 y)$]

#solutionbox[
#math.equation(block: true, alt: "−5 y minus 3 plus the fraction 1 over 4 y")[$−5 y − 3 + frac(1, 4 y)$]
]

#strong[Divide Polynomials using Long Division]

In the following exercises, divide each polynomial by the binomial.

#math.equation(block: true, alt: "open parenthesis y squared plus 7 y plus 12 close parenthesis divided by open parenthesis y plus 3 close parenthesis")[$( y^(2) + 7 y + 12 ) ÷ ( y + 3 )$]

#math.equation(block: true, alt: "open parenthesis a squared minus 2 a minus 35 close parenthesis divided by open parenthesis a plus 5 close parenthesis")[$( a^(2) − 2 a − 35 ) ÷ ( a + 5 )$]

#solutionbox[
#math.equation(block: true, alt: "a minus 7")[$a − 7$]
]

#math.equation(block: true, alt: "open parenthesis 6 m squared minus 19 m minus 20 close parenthesis divided by open parenthesis m minus 4 close parenthesis")[$( 6 m^(2) − 19 m − 20 ) ÷ ( m − 4 )$]

#math.equation(block: true, alt: "open parenthesis 4 x squared minus 17 x minus 15 close parenthesis divided by open parenthesis x minus 5 close parenthesis")[$( 4 x^(2) − 17 x − 15 ) ÷ ( x − 5 )$]

#solutionbox[
#math.equation(block: true, alt: "4 x plus 3")[$4 x + 3$]
]

#math.equation(block: true, alt: "open parenthesis q squared plus 2 q plus 20 close parenthesis divided by open parenthesis q plus 6 close parenthesis")[$( q^(2) + 2 q + 20 ) ÷ ( q + 6 )$]

#math.equation(block: true, alt: "open parenthesis p squared plus 11 p plus 16 close parenthesis divided by open parenthesis p plus 8 close parenthesis")[$( p^(2) + 11 p + 16 ) ÷ ( p + 8 )$]

#solutionbox[
#math.equation(block: true, alt: "p plus 3 minus the fraction 8 over p plus 8")[$p + 3 − frac(8, p + 8)$]
]

#math.equation(block: true, alt: "open parenthesis 3 b cubed plus b squared plus 4 close parenthesis divided by open parenthesis b plus 1 close parenthesis")[$( 3 b^(3) + b^(2) + 4 ) ÷ ( b + 1 )$]

#math.equation(block: true, alt: "open parenthesis 2 n cubed minus 10 n plus 28 close parenthesis divided by open parenthesis n plus 3 close parenthesis")[$( 2 n^(3) − 10 n + 28 ) ÷ ( n + 3 )$]

#solutionbox[
#math.equation(block: true, alt: "2 n squared minus 6 n plus 8 plus the fraction 4 over n plus 3")[$2 n^(2) − 6 n + 8 + frac(4, n + 3)$]
]

#math.equation(block: true, alt: "open parenthesis z cubed plus 1 close parenthesis divided by open parenthesis z plus 1 close parenthesis")[$( z^(3) + 1 ) ÷ ( z + 1 )$]

#math.equation(block: true, alt: "open parenthesis m cubed plus 1000 close parenthesis divided by open parenthesis m plus 10 close parenthesis")[$( m^(3) + 1000 ) ÷ ( m + 10 )$]

#solutionbox[
#math.equation(block: true, alt: "m squared minus 10 m plus 100")[$m^(2) − 10 m + 100$]
]

#math.equation(block: true, alt: "open parenthesis 64 x cubed minus 27 close parenthesis divided by open parenthesis 4 x minus 3 close parenthesis")[$( 64 x^(3) − 27 ) ÷ ( 4 x − 3 )$]

#math.equation(block: true, alt: "open parenthesis 125 y cubed minus 64 close parenthesis divided by open parenthesis 5 y minus 4 close parenthesis")[$( 125 y^(3) − 64 ) ÷ ( 5 y − 4 )$]

#solutionbox[
#math.equation(block: true, alt: "25 y squared plus 20 y plus 16")[$25 y^(2) + 20 y + 16$]
]

#strong[Divide Polynomials using Synthetic Division]

In the following exercises, use synthetic Division to find the quotient and remainder.

#math.equation(block: false, alt: "x cubed minus 6 x squared plus 5 x plus 14")[$x^(3) − 6 x^(2) + 5 x + 14$] is divided by #math.equation(block: false, alt: "x plus 1")[$x + 1$]

#math.equation(block: false, alt: "x cubed minus 3 x squared minus 4 x plus 12")[$x^(3) − 3 x^(2) − 4 x + 12$] is divided by #math.equation(block: false, alt: "x plus 2")[$x + 2$]

#solutionbox[
#math.equation(block: true, alt: "x squared minus 5 x plus 6 ; 0")[$x^(2) − 5 x + 6 ";" #h(0.2em) 0$]
]

#math.equation(block: false, alt: "2 x cubed minus 11 x squared plus 11 x plus 12")[$2 x^(3) − 11 x^(2) + 11 x + 12$] is divided by #math.equation(block: false, alt: "x minus 3")[$x − 3$]

#math.equation(block: false, alt: "2 x cubed minus 11 x squared plus 16 x minus 12")[$2 x^(3) − 11 x^(2) + 16 x − 12$] is divided by #math.equation(block: false, alt: "x minus 4")[$x − 4$]

#solutionbox[
#math.equation(block: true, alt: "2 x squared minus 3 x plus 4 ; 4")[$2 x^(2) − 3 x + 4 ";" #h(0.2em) 4$]
]

#math.equation(block: false, alt: "x to the power 4 minus 5 x squared plus 13 x plus 3")[$x^(4) − 5 x^(2) + 13 x + 3$] is divided by #math.equation(block: false, alt: "x plus 3")[$x + 3$]

#math.equation(block: false, alt: "x to the power 4 plus x squared plus 6 x minus 10")[$x^(4) + x^(2) + 6 x − 10$] is divided by #math.equation(block: false, alt: "x plus 2")[$x + 2$]

#solutionbox[
#math.equation(block: true, alt: "x cubed minus 2 x squared plus 5 x minus 4 ; − 2")[$x^(3) − 2 x^(2) + 5 x − 4 ";" #h(0.2em) "−" 2$]
]

#math.equation(block: false, alt: "2 x to the power 4 minus 9 x cubed plus 5 x squared minus 3 x minus 6")[$2 x^(4) − 9 x^(3) + 5 x^(2) − 3 x − 6$] is divided by #math.equation(block: false, alt: "x minus 4")[$x − 4$]

#math.equation(block: false, alt: "3 x to the power 4 minus 11 x cubed plus 2 x squared plus 10 x plus 6")[$3 x^(4) − 11 x^(3) + 2 x^(2) + 10 x + 6$] is divided by #math.equation(block: false, alt: "x minus 3")[$x − 3$]

#solutionbox[
#math.equation(block: true, alt: "3 x cubed minus 2 x squared minus 4 x minus 2 ; 0")[$3 x^(3) − 2 x^(2) − 4 x − 2 ";" #h(0.2em) 0$]
]

#strong[Divide Polynomial Functions]

In the following exercises, divide.

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 13 x plus 36")[$f ( x ) = x^(2) − 13 x + 36$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x minus 4 ,")[$g ( x ) = x − 4 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −1 close parenthesis")[$( frac(f, g) ) ( −1 )$]

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 15 x plus 54")[$f ( x ) = x^(2) − 15 x + 54$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x minus 9 ,")[$g ( x ) = x − 9 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −5 close parenthesis")[$( frac(f, g) ) ( −5 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals x minus 6")[$( frac(f, g) ) ( x ) = x − 6$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −5 close parenthesis equals −11")[$( frac(f, g) ) ( −5 ) = −11$]
]

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus x squared minus 7 x plus 2")[$f ( x ) = x^(3) + x^(2) − 7 x + 2$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x minus 2 ,")[$g ( x ) = x − 2 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 2 close parenthesis")[$( frac(f, g) ) ( 2 )$]

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus 2 x squared minus 19 x plus 12")[$f ( x ) = x^(3) + 2 x^(2) − 19 x + 12$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x minus 3 ,")[$g ( x ) = x − 3 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 0 close parenthesis")[$( frac(f, g) ) ( 0 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals x squared plus 5 x minus 4")[$( frac(f, g) ) ( x ) = x^(2) + 5 x − 4$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 0 close parenthesis equals −4")[$( frac(f, g) ) ( 0 ) = −4$]
]

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 3 x plus 2")[$f ( x ) = x^(2) − 3 x + 2$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x plus 3 ,")[$g ( x ) = x + 3 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 3 close parenthesis")[$( frac(f, g) ) ( 3 )$]

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared plus 2 x minus 3")[$f ( x ) = x^(2) + 2 x − 3$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x plus 3")[$g ( x ) = x + 3$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$]ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 3 close parenthesis")[$( frac(f, g) ) ( 3 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction f over g x equals x minus 1 ; x not equal to minus 3")[$frac(f, g) x = x − 1 ; x ≠ − 3$] 
  ⓑ #math.equation(block: false, alt: "the fraction f over g 3 equals 2")[$frac(f, g) 3 = 2$]
]

#strong[Use the Remainder and Factor Theorem]

In the following exercises, use the Remainder Theorem to find the remainder.

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 8 x plus 7")[$f ( x ) = x^(3) − 8 x + 7$] is divided by #math.equation(block: false, alt: "x plus 3")[$x + 3$]

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 4 x minus 9")[$f ( x ) = x^(3) − 4 x − 9$] is divided by #math.equation(block: false, alt: "x plus 2")[$x + 2$]

#solutionbox[
#math.equation(block: true, alt: "−9")[$−9$]
]

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x cubed minus 6 x minus 24")[$f ( x ) = 2 x^(3) − 6 x − 24$] divided by #math.equation(block: false, alt: "x minus 3")[$x − 3$]

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 7 x squared minus 5 x minus 8")[$f ( x ) = 7 x^(2) − 5 x − 8$] divided by #math.equation(block: false, alt: "x minus 1")[$x − 1$]

#solutionbox[
−6
]

In the following exercises, use the Factor Theorem to determine if #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of the polynomial function.

Determine whether #math.equation(block: false, alt: "x plus 3")[$x + 3$] a factor of #math.equation(block: false, alt: "x cubed plus 8 x squared plus 21 x plus 18")[$x^(3) + 8 x^(2) + 21 x + 18$]

Determine whether #math.equation(block: false, alt: "x plus 4")[$x + 4$] a factor of #math.equation(block: false, alt: "x cubed plus x squared minus 14 x plus 8")[$x^(3) + x^(2) − 14 x + 8$]

#solutionbox[
no
]

Determine whether #math.equation(block: false, alt: "x minus 2")[$x − 2$] a factor of #math.equation(block: false, alt: "x cubed minus 7 x squared plus 7 x minus 6")[$x^(3) − 7 x^(2) + 7 x − 6$]

Determine whether #math.equation(block: false, alt: "x minus 3")[$x − 3$] a factor of #math.equation(block: false, alt: "x cubed minus 7 x squared plus 11 x plus 3")[$x^(3) − 7 x^(2) + 11 x + 3$]

#solutionbox[
yes
]


==== Writing Exercises

James divides #math.equation(block: false, alt: "48 y plus 6")[$48 y + 6$] by 6 this way: #math.equation(block: false, alt: "the fraction 48 y plus 6 over 6 equals 48 y .")[$frac(48 y + cancel(6), cancel(6)) = 48 y .$] What is wrong with his reasoning?

Divide #math.equation(block: false, alt: "the fraction 10 x squared plus x minus 12 over 2 x")[$frac(10 x^(2) + x − 12, 2 x)$] and explain with words how you get each term of the quotient.

#solutionbox[
answer will vary
]

Explain when you can use synthetic division.

In your own words, write the steps for synthetic division for #math.equation(block: false, alt: "x squared plus 5 x plus 6")[$x^(2) + 5 x + 6$] divided by #math.equation(block: false, alt: "x minus 2 .")[$x − 2 .$]

#solutionbox[
Answers will vary.
]


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section

#figure(figph[The figure shows a table with seven rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “divide monomials”, “divide a polynomial by using a monomial”, “divide polynomials using long division”, “divide polynomials using synthetic division”, “divide polynomial functions”, and “use the Remainder and Factor Theorem”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.], alt: "The figure shows a table with seven rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is \"confidently\", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “divide monomials”, “divide a polynomial by using a monomial”, “divide polynomials using long division”, “divide polynomials using synthetic division”, “divide polynomial functions”, and “use the Remainder and Factor Theorem”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.", caption: none)

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


=== Chapter Review Exercises


==== Add and Subtract Polynomials

#strong[Types of Polynomials]

In the following exercises, determine the type of polynomial.

#math.equation(block: true, alt: "16 x squared minus 40 x minus 25")[$16 x^(2) − 40 x − 25$]

#math.equation(block: true, alt: "5 m plus 9")[$5 m + 9$]

#solutionbox[
binomial
]

#math.equation(block: true, alt: "−15")[$−15$]

#math.equation(block: true, alt: "y squared plus 6 y cubed plus 9 y to the power 4")[$y^(2) + 6 y^(3) + 9 y^(4)$]

#solutionbox[
trinomial
]

#strong[Add and Subtract Polynomials]

In the following exercises, add or subtract the polynomials.

#math.equation(block: true, alt: "4 p plus 11 p")[$4 p + 11 p$]

#math.equation(block: true, alt: "−8 y cubed minus 5 y cubed")[$−8 y^(3) − 5 y^(3)$]

#solutionbox[
#math.equation(block: true, alt: "−13 y cubed")[$−13 y^(3)$]
]

#math.equation(block: true, alt: "open parenthesis 4 a squared plus 9 a minus 11 close parenthesis plus open parenthesis 6 a squared minus 5 a plus 10 close parenthesis")[$( 4 a^(2) + 9 a − 11 ) + ( 6 a^(2) − 5 a + 10 )$]

#math.equation(block: true, alt: "open parenthesis 8 m squared plus 12 m minus 5 close parenthesis minus open parenthesis 2 m squared minus 7 m minus 1 close parenthesis")[$( 8 m^(2) + 12 m − 5 ) − ( 2 m^(2) − 7 m − 1 )$]

#solutionbox[
#math.equation(block: true, alt: "6 m squared plus 19 m minus 4")[$6 m^(2) + 19 m − 4$]
]

#math.equation(block: true, alt: "open parenthesis y squared minus 3 y plus 12 close parenthesis plus open parenthesis 5 y squared minus 9 close parenthesis")[$( y^(2) − 3 y + 12 ) + ( 5 y^(2) − 9 )$]

#math.equation(block: true, alt: "open parenthesis 5 u squared plus 8 u close parenthesis minus open parenthesis 4 u minus 7 close parenthesis")[$( 5 u^(2) + 8 u ) − ( 4 u − 7 )$]

#solutionbox[
#math.equation(block: true, alt: "5 u squared plus 4 u plus 7")[$5 u^(2) + 4 u + 7$]
]

Find the sum of #math.equation(block: false, alt: "8 q cubed minus 27")[$8 q^(3) − 27$] and #math.equation(block: false, alt: "q squared plus 6 q minus 2 .")[$q^(2) + 6 q − 2 .$]

Find the difference of #math.equation(block: false, alt: "x squared plus 6 x plus 8")[$x^(2) + 6 x + 8$] and #math.equation(block: false, alt: "x squared minus 8 x plus 15 .")[$x^(2) − 8 x + 15 .$]

#solutionbox[
#math.equation(block: true, alt: "14 x minus 7")[$14 x − 7$]
]

In the following exercises, simplify.

#math.equation(block: true, alt: "17 m n squared minus open parenthesis −9 m n squared close parenthesis plus 3 m n squared")[$17 m n^(2) − ( −9 m n^(2) ) + 3 m n^(2)$]

#math.equation(block: true, alt: "18 a minus 7 b minus 21 a")[$18 a − 7 b − 21 a$]

#solutionbox[
#math.equation(block: true, alt: "−7 b minus 3 a")[$−7 b − 3 a$]
]

#math.equation(block: true, alt: "2 p q squared minus 5 p minus 3 q squared")[$2 p q^(2) − 5 p − 3 q^(2)$]

#math.equation(block: true, alt: "open parenthesis 6 a squared plus 7 close parenthesis plus open parenthesis 2 a squared minus 5 a minus 9 close parenthesis")[$( 6 a^(2) + 7 ) + ( 2 a^(2) − 5 a − 9 )$]

#solutionbox[
#math.equation(block: true, alt: "8 a squared minus 5 a minus 2")[$8 a^(2) − 5 a − 2$]
]

#math.equation(block: true, alt: "open parenthesis 3 p squared minus 4 p minus 9 close parenthesis plus open parenthesis 5 p squared plus 14 close parenthesis")[$( 3 p^(2) − 4 p − 9 ) + ( 5 p^(2) + 14 )$]

#math.equation(block: true, alt: "open parenthesis 7 m squared minus 2 m minus 5 close parenthesis minus open parenthesis 4 m squared plus m minus 8 close parenthesis")[$( 7 m^(2) − 2 m − 5 ) − ( 4 m^(2) + m − 8 )$]

#solutionbox[
#math.equation(block: true, alt: "3 m squared minus 3 m plus 3")[$3 m^(2) − 3 m + 3$]
]

#math.equation(block: true, alt: "open parenthesis 7 b squared minus 4 b plus 3 close parenthesis minus open parenthesis 8 b squared minus 5 b minus 7 close parenthesis")[$( 7 b^(2) − 4 b + 3 ) − ( 8 b^(2) − 5 b − 7 )$]

Subtract #math.equation(block: false, alt: "open parenthesis 8 y squared minus y plus 9 close parenthesis")[$( 8 y^(2) − y + 9 )$] from #math.equation(block: false, alt: "open parenthesis 11 y squared minus 9 y minus 5 close parenthesis")[$( 11 y^(2) − 9 y − 5 )$]

#solutionbox[
#math.equation(block: true, alt: "3 y squared minus 8 y minus 14")[$3 y^(2) − 8 y − 14$]
]

Find the difference of #math.equation(block: false, alt: "open parenthesis z squared minus 4 z minus 12 close parenthesis")[$( z^(2) − 4 z − 12 )$] and #math.equation(block: false, alt: "open parenthesis 3 z squared plus 2 z minus 11 close parenthesis")[$( 3 z^(2) + 2 z − 11 )$]

#math.equation(block: true, alt: "open parenthesis x cubed minus x squared y close parenthesis minus open parenthesis 4 x y squared minus y cubed close parenthesis plus open parenthesis 3 x squared y minus x y squared close parenthesis")[$( x^(3) − x^(2) y ) − ( 4 x y^(2) − y^(3) ) + ( 3 x^(2) y − x y^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "x cubed plus 2 x squared y minus 5 x y squared plus y cubed")[$x^(3) + 2 x^(2) y − 5 x y^(2) + y^(3)$]
]

#math.equation(block: true, alt: "open parenthesis x cubed minus 2 x squared y close parenthesis minus open parenthesis x y squared minus 3 y cubed close parenthesis minus open parenthesis x squared y minus 4 x y squared close parenthesis")[$( x^(3) − 2 x^(2) y ) − ( x y^(2) − 3 y^(3) ) − ( x^(2) y − 4 x y^(2) )$]

#strong[Evaluate a Polynomial Function for a Given Value of the Variable]

In the following exercises, find the function values for each polynomial function.

For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 7 x squared minus 3 x plus 5")[$f ( x ) = 7 x^(2) − 3 x + 5$] find: #linebreak() ⓐ #math.equation(block: false, alt: "f open parenthesis 5 close parenthesis")[$f ( 5 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis")[$f ( −2 )$] ⓒ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis")[$f ( 0 )$]

#solutionbox[
ⓐ 165 ⓑ 39 ⓒ 5
]

For the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 15 minus 16 x squared ,")[$g ( x ) = 15 − 16 x^(2) ,$] find: #linebreak() ⓐ #math.equation(block: false, alt: "g open parenthesis −1 close parenthesis")[$g ( −1 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis 0 close parenthesis")[$g ( 0 )$] ⓒ #math.equation(block: false, alt: "g open parenthesis 2 close parenthesis")[$g ( 2 )$]

A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function #math.equation(block: false, alt: "h open parenthesis t close parenthesis equals −16 t squared plus 640")[$h ( t ) = −16 t^(2) + 640$] gives the height of the glasses #emph[t] seconds after they were dropped. Find the height of the glasses when #math.equation(block: false, alt: "t equals 6 .")[$t = 6 .$]

#solutionbox[
The height is #math.equation(block: false, alt: "64 feet")[$64 #h(0.2em) "feet"$].
]

A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of #math.equation(block: false, alt: "p")[$p$] dollars each is given by the polynomial #math.equation(block: false, alt: "R open parenthesis p close parenthesis equals −5 p squared plus 360 p .")[$R ( p ) = −5 p^(2) + 360 p .$] Find the revenue received when #math.equation(block: false, alt: "p equals 10")[$p = 10$] dollars.

#strong[Add and Subtract Polynomial Functions]

In the following exercises, find ⓐ (#emph[f] + #emph[g])(#emph[x]) ⓑ (#emph[f] + #emph[g])(3) ⓒ (#emph[f] − #emph[g])(#emph[x]) ⓓ (#emph[f] − #emph[g])(−2)

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x squared minus 4 x minus 7")[$f ( x ) = 2 x^(2) − 4 x − 7$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 x squared minus x plus 5")[$g ( x ) = 2 x^(2) − x + 5$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis f plus g close parenthesis open parenthesis x close parenthesis equals 4 x squared minus 5 x minus 2")[$( f + g ) ( x ) = 4 x^(2) − 5 x − 2$] ⓑ #math.equation(block: false, alt: "open parenthesis f plus g close parenthesis open parenthesis 3 close parenthesis equals 19")[$( f + g ) ( 3 ) = 19$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis f minus g close parenthesis open parenthesis x close parenthesis equals −3 x minus 12")[$( f − g ) ( x ) = −3 x − 12$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis f minus g close parenthesis open parenthesis −2 close parenthesis equals −6")[$( f − g ) ( −2 ) = −6$]
]

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 4 x cubed minus 3 x squared plus x minus 1")[$f ( x ) = 4 x^(3) − 3 x^(2) + x − 1$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 8 x cubed minus 1")[$g ( x ) = 8 x^(3) − 1$]


==== Properties of Exponents and Scientific Notation

#strong[Simplify Expressions Using the Properties for Exponents]

In the following exercises, simplify each expression using the properties for exponents.

#math.equation(block: true, alt: "p cubed times p to the power 10")[$p^(3) · p^(10)$]

#solutionbox[
#math.equation(block: true, alt: "p to the power 13")[$p^(13)$]
]

#math.equation(block: true, alt: "2 times 2 to the power 6")[$2 · 2^(6)$]

#math.equation(block: true, alt: "a times a squared times a cubed")[$a · a^(2) · a^(3)$]

#solutionbox[
#math.equation(block: true, alt: "a to the power 6")[$a^(6)$]
]

#math.equation(block: true, alt: "x times x to the power 8")[$x · x^(8)$]

#math.equation(block: true, alt: "y to the power a times y to the power b")[$y^(a) · y^(b)$]

#solutionbox[
#math.equation(block: true, alt: "y to the power a plus b")[$y^(a + b)$]
]

#math.equation(block: true, alt: "the fraction 2 to the power 8 over 2 squared")[$frac(2^(8), 2^(2))$]

#math.equation(block: true, alt: "the fraction a to the power 6 over a")[$frac(a^(6), a)$]

#solutionbox[
#math.equation(block: true, alt: "a to the power 5")[$a^(5)$]
]

#math.equation(block: true, alt: "the fraction n cubed over n to the power 12")[$frac(n^(3), n^(12))$]

#math.equation(block: true, alt: "the fraction 1 over x to the power 5")[$frac(1, x^(5))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over x to the power 5")[$frac(1, x^(5))$]
]

#math.equation(block: true, alt: "3 to the power 0")[$3^(0)$]

#math.equation(block: true, alt: "y to the power 0")[$y^(0)$]

#solutionbox[
1
]

#math.equation(block: true, alt: "open parenthesis 14 t close parenthesis to the power 0")[$attach(( 14 t ), t: 0)$]

#math.equation(block: true, alt: "12 a to the power 0 minus 15 b to the power 0")[$12 a^(0) − 15 b^(0)$]

#solutionbox[
#math.equation(block: true, alt: "−3")[$−3$]
]

#strong[Use the Definition of a Negative Exponent]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "6 to the power −2")[$6^(−2)$]

#math.equation(block: true, alt: "open parenthesis −10 close parenthesis to the power −3")[$attach(( −10 ), t: −3)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 1000")[$− frac(1, 1000)$]
]

#math.equation(block: true, alt: "5 times 2 to the power −4")[$5 · 2^(−4)$]

#math.equation(block: true, alt: "open parenthesis 8 n close parenthesis to the power −1")[$attach(( 8 n ), t: −1)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 8 n")[$frac(1, 8 n)$]
]

#math.equation(block: true, alt: "y to the power −5")[$y^(−5)$]

#math.equation(block: true, alt: "10 to the power −3")[$10^(−3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 1000")[$frac(1, 1000)$]
]

#math.equation(block: true, alt: "the fraction 1 over a to the power −4")[$frac(1, a^(−4))$]

#math.equation(block: true, alt: "the fraction 1 over 6 to the power −2")[$frac(1, 6^(−2))$]

#solutionbox[
36
]

#math.equation(block: true, alt: "− 5 to the power −3")[$"−" 5^(−3)$]

#math.equation(block: true, alt: "open parenthesis minus the fraction 1 over 5 close parenthesis to the power −3")[$attach(( − frac(1, 5) ), t: −3)$]

#solutionbox[
#math.equation(block: true, alt: "−125")[$−125$]
]

#math.equation(block: true, alt: "− open parenthesis the fraction 1 over 2 close parenthesis to the power −3")[$"−" attach(( frac(1, 2) ), t: −3)$]

#math.equation(block: true, alt: "open parenthesis −5 close parenthesis to the power −3")[$attach(( −5 ), t: −3)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 125")[$− frac(1, 125)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 5 over 9 close parenthesis to the power −2")[$attach(( frac(5, 9) ), t: −2)$]

#math.equation(block: true, alt: "open parenthesis minus the fraction 3 over x close parenthesis to the power −3")[$attach(( − frac(3, x) ), t: −3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction minus x cubed over 27")[$frac(attach(− x, t: 3), 27)$]
]

In the following exercises, simplify each expression using the Product Property.

#math.equation(block: true, alt: "open parenthesis y to the power 4 close parenthesis cubed")[$attach(( y^(4) ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis 3 squared close parenthesis to the power 5")[$attach(( 3^(2) ), t: 5)$]

#solutionbox[
#math.equation(block: true, alt: "3 to the power 10")[$3^(10)$]
]

#math.equation(block: true, alt: "open parenthesis a to the power 10 close parenthesis to the power y")[$attach(( a^(10) ), t: y)$]

#math.equation(block: true, alt: "x to the power −3 times x to the power 9")[$x^(−3) · x^(9)$]

#solutionbox[
#math.equation(block: true, alt: "x to the power 6")[$x^(6)$]
]

#math.equation(block: true, alt: "r to the power −5 times r to the power −4")[$r^(−5) · r^(−4)$]

#math.equation(block: true, alt: "open parenthesis u v to the power −3 close parenthesis open parenthesis u to the power −4 v to the power −2 close parenthesis")[$( u v^(−3) ) ( u^(−4) v^(−2) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over u cubed v to the power 5")[$frac(1, u^(3) v^(5))$]
]

#math.equation(block: true, alt: "open parenthesis m to the power 5 close parenthesis to the power −1")[$attach(( m^(5) ), t: −1)$]

#math.equation(block: true, alt: "p to the power 5 times p to the power −2 times p to the power −4")[$p^(5) · p^(−2) · p^(−4)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over p")[$frac(1, p)$]
]

In the following exercises, simplify each expression using the Power Property.

#math.equation(block: true, alt: "open parenthesis k to the power −2 close parenthesis to the power −3")[$attach(( k^(−2) ), t: −3)$]

#math.equation(block: true, alt: "the fraction q to the power 4 over q to the power 20")[$frac(q^(4), q^(20))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over q to the power 16")[$frac(1, q^(16))$]
]

#math.equation(block: true, alt: "the fraction b to the power 8 over b to the power −2")[$frac(b^(8), b^(−2))$]

#math.equation(block: true, alt: "the fraction n to the power −3 over n to the power −5")[$frac(n^(−3), n^(−5))$]

#solutionbox[
#math.equation(block: true, alt: "n squared")[$n^(2)$]
]

In the following exercises, simplify each expression using the Product to a Power Property.

#math.equation(block: true, alt: "open parenthesis −5 a b close parenthesis cubed")[$attach(( −5 a b ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis −4 p q close parenthesis to the power 0")[$attach(( −4 p q ), t: 0)$]

#solutionbox[
1
]

#math.equation(block: true, alt: "open parenthesis −6 x cubed close parenthesis to the power −2")[$attach(( −6 x^(3) ), t: −2)$]

#math.equation(block: true, alt: "open parenthesis 3 y to the power −4 close parenthesis squared")[$attach(( 3 y^(−4) ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 over y to the power 8")[$frac(9, y^(8))$]
]

In the following exercises, simplify each expression using the Quotient to a Power Property.

#math.equation(block: true, alt: "open parenthesis the fraction 3 over 5 x close parenthesis to the power −2")[$attach(( frac(3, 5 x) ), t: −2)$]

#math.equation(block: true, alt: "open parenthesis the fraction 3 x y squared over z close parenthesis to the power 4")[$attach(( frac(3 x y^(2), z) ), t: 4)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 81 x to the power 4 y to the power 8 over z to the power 4")[$frac(81 x^(4) y^(8), z^(4))$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 4 p to the power −3 over q squared close parenthesis squared")[$attach(( frac(4 p^(−3), q^(2)) ), t: 2)$]

In the following exercises, simplify each expression by applying several properties.

#math.equation(block: true, alt: "open parenthesis x squared y close parenthesis squared open parenthesis 3 x y to the power 5 close parenthesis cubed")[$attach(( x^(2) y ), t: 2) attach(( 3 x y^(5) ), t: 3)$]

#solutionbox[
#math.equation(block: true, alt: "27 x to the power 7 y to the power 17")[$27 x^(7) y^(17)$]
]

#math.equation(block: true, alt: "the fraction open parenthesis −3 a to the power −2 close parenthesis to the power 4 open parenthesis 2 a to the power 4 close parenthesis squared over open parenthesis −6 a squared close parenthesis cubed")[$frac(attach(( −3 a^(−2) ), t: 4) attach(( 2 a^(4) ), t: 2), attach(( −6 a^(2) ), t: 3))$]

#math.equation(block: true, alt: "open parenthesis the fraction 3 x y cubed over 4 x to the power 4 y to the power −2 close parenthesis squared open parenthesis the fraction 6 x y to the power 4 over 8 x cubed y to the power −2 close parenthesis to the power −1")[$attach(( frac(3 x y^(3), 4 x^(4) y^(−2)) ), t: 2) attach(( frac(6 x y^(4), 8 x^(3) y^(−2)) ), t: −1)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 y to the power 4 over 4 x to the power 4")[$frac(3 y^(4), 4 x^(4))$]
]

In the following exercises, write each number in scientific notation.

#math.equation(block: true, alt: "2.568")[$2.568$]

5,300,000

#solutionbox[
#math.equation(block: true, alt: "5.3 times 10 to the power 6")[$5.3 #h(0.2em) × #h(0.2em) 10^(6)$]
]

#math.equation(block: true, alt: "0.00814")[$0.00814$]

In the following exercises, convert each number to decimal form.

#math.equation(block: true, alt: "2.9 times 10 to the power 4")[$2.9 #h(0.2em) × #h(0.2em) 10^(4)$]

#solutionbox[
#math.equation(block: true, alt: "29,000")[$29,000$]
]

#math.equation(block: true, alt: "3.75 times 10 to the power −1")[$3.75 #h(0.2em) × #h(0.2em) 10^(−1)$]

#math.equation(block: true, alt: "9.413 times 10 to the power −5")[$9.413 #h(0.2em) × #h(0.2em) 10^(−5)$]

#solutionbox[
#math.equation(block: true, alt: "0.00009413")[$0.00009413$]
]

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

#math.equation(block: true, alt: "open parenthesis 3 times 10 to the power 7 close parenthesis open parenthesis 2 times 10 to the power −4 close parenthesis")[$( 3 #h(0.2em) × #h(0.2em) 10^(7) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−4) )$]

#math.equation(block: true, alt: "open parenthesis 1.5 times 10 to the power −3 close parenthesis open parenthesis 4.8 times 10 to the power −1 close parenthesis")[$( 1.5 #h(0.2em) × #h(0.2em) 10^(−3) ) ( 4.8 #h(0.2em) × #h(0.2em) 10^(−1) )$]

#solutionbox[
#math.equation(block: true, alt: "0.00072")[$0.00072$]
]

#math.equation(block: true, alt: "the fraction 6 times 10 to the power 9 over 2 times 10 to the power −1")[$frac(6 #h(0.2em) × #h(0.2em) 10^(9), 2 #h(0.2em) × #h(0.2em) 10^(−1))$]

#math.equation(block: true, alt: "the fraction 9 times 10 to the power −3 over 1 times 10 to the power −6")[$frac(9 #h(0.2em) × #h(0.2em) 10^(−3), 1 #h(0.2em) × #h(0.2em) 10^(−6))$]

#solutionbox[
#math.equation(block: true, alt: "9,000")[$9,000$]
]


==== Multiply Polynomials

#strong[Multiply Monomials]

In the following exercises, multiply the monomials.

#math.equation(block: true, alt: "open parenthesis −6 p to the power 4 close parenthesis open parenthesis 9 p close parenthesis")[$( −6 p^(4) ) ( 9 p )$]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 3 c squared close parenthesis open parenthesis 30 c to the power 8 close parenthesis")[$( frac(1, 3) c^(2) ) ( 30 c^(8) )$]

#solutionbox[
#math.equation(block: true, alt: "10 c to the power 10")[$10 c^(10)$]
]

#math.equation(block: true, alt: "open parenthesis 8 x squared y to the power 5 close parenthesis open parenthesis 7 x y to the power 6 close parenthesis")[$( 8 x^(2) y^(5) ) ( 7 x y^(6) )$]

#math.equation(block: true, alt: "open parenthesis the fraction 2 over 3 m cubed n to the power 6 close parenthesis open parenthesis the fraction 1 over 6 m to the power 4 n to the power 4 close parenthesis")[$( frac(2, 3) m^(3) n^(6) ) ( frac(1, 6) m^(4) n^(4) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction m to the power 7 n to the power 10 over 9")[$frac(m^(7) n^(10), 9)$]
]

#strong[Multiply a Polynomial by a Monomial]

In the following exercises, multiply.

#math.equation(block: true, alt: "7 open parenthesis 10 minus x close parenthesis")[$7 ( 10 − x )$]

#math.equation(block: true, alt: "a squared open parenthesis a squared minus 9 a minus 36 close parenthesis")[$a^(2) ( a^(2) − 9 a − 36 )$]

#solutionbox[
#math.equation(block: true, alt: "a to the power 4 minus 9 a cubed minus 36 a squared")[$a^(4) − 9 a^(3) − 36 a^(2)$]
]

#math.equation(block: true, alt: "−5 y open parenthesis 125 y cubed minus 1 close parenthesis")[$−5 y ( 125 y^(3) − 1 )$]

#math.equation(block: true, alt: "open parenthesis 4 n minus 5 close parenthesis open parenthesis 2 n cubed close parenthesis")[$( 4 n − 5 ) ( 2 n^(3) )$]

#solutionbox[
#math.equation(block: true, alt: "8 n to the power 4 minus 10 n cubed")[$8 n^(4) − 10 n^(3)$]
]

#strong[Multiply a Binomial by a Binomial]

In the following exercises, multiply the binomials using:

ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

#math.equation(block: true, alt: "open parenthesis a plus 5 close parenthesis open parenthesis a plus 2 close parenthesis")[$( a + 5 ) ( a + 2 )$]

#math.equation(block: true, alt: "open parenthesis y minus 4 close parenthesis open parenthesis y plus 12 close parenthesis")[$( y − 4 ) ( y + 12 )$]

#solutionbox[
#math.equation(block: true, alt: "y squared plus 8 y minus 48")[$y^(2) + 8 y − 48$]
]

#math.equation(block: true, alt: "open parenthesis 3 x plus 1 close parenthesis open parenthesis 2 x minus 7 close parenthesis")[$( 3 x + 1 ) ( 2 x − 7 )$]

#math.equation(block: true, alt: "open parenthesis 6 p minus 11 close parenthesis open parenthesis 3 p minus 10 close parenthesis")[$( 6 p − 11 ) ( 3 p − 10 )$]

#solutionbox[
#math.equation(block: true, alt: "18 p squared minus 93 p plus 110")[$18 p^(2) − 93 p + 110$]
]

In the following exercises, multiply the binomials. Use any method.

#math.equation(block: true, alt: "open parenthesis n plus 8 close parenthesis open parenthesis n plus 1 close parenthesis")[$( n + 8 ) ( n + 1 )$]

#math.equation(block: true, alt: "open parenthesis k plus 6 close parenthesis open parenthesis k minus 9 close parenthesis")[$( k + 6 ) ( k − 9 )$]

#solutionbox[
#math.equation(block: true, alt: "k squared minus 3 k minus 54")[$k^(2) − 3 k − 54$]
]

#math.equation(block: true, alt: "open parenthesis 5 u minus 3 close parenthesis open parenthesis u plus 8 close parenthesis")[$( 5 u − 3 ) ( u + 8 )$]

#math.equation(block: true, alt: "open parenthesis 2 y minus 9 close parenthesis open parenthesis 5 y minus 7 close parenthesis")[$( 2 y − 9 ) ( 5 y − 7 )$]

#solutionbox[
#math.equation(block: true, alt: "10 y squared minus 59 y plus 63")[$10 y^(2) − 59 y + 63$]
]

#math.equation(block: true, alt: "open parenthesis p plus 4 close parenthesis open parenthesis p plus 7 close parenthesis")[$( p + 4 ) ( p + 7 )$]

#math.equation(block: true, alt: "open parenthesis x minus 8 close parenthesis open parenthesis x plus 9 close parenthesis")[$( x − 8 ) ( x + 9 )$]

#solutionbox[
#math.equation(block: true, alt: "x squared plus x minus 72")[$x^(2) + x − 72$]
]

#math.equation(block: true, alt: "open parenthesis 3 c plus 1 close parenthesis open parenthesis 9 c minus 4 close parenthesis")[$( 3 c + 1 ) ( 9 c − 4 )$]

#math.equation(block: true, alt: "open parenthesis 10 a minus 1 close parenthesis open parenthesis 3 a minus 3 close parenthesis")[$( 10 a − 1 ) ( 3 a − 3 )$]

#solutionbox[
#math.equation(block: true, alt: "30 a squared minus 33 a plus 3")[$30 a^(2) − 33 a + 3$]
]

#strong[Multiply a Polynomial by a Polynomial]

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

#math.equation(block: true, alt: "open parenthesis x plus 1 close parenthesis open parenthesis x squared minus 3 x minus 21 close parenthesis")[$( x + 1 ) ( x^(2) − 3 x − 21 )$]

#math.equation(block: true, alt: "open parenthesis 5 b minus 2 close parenthesis open parenthesis 3 b squared plus b minus 9 close parenthesis")[$( 5 b − 2 ) ( 3 b^(2) + b − 9 )$]

#solutionbox[
#math.equation(block: true, alt: "15 b cubed minus b squared minus 47 b plus 18")[$15 b^(3) − b^(2) − 47 b + 18$]
]

In the following exercises, multiply. Use either method.

#math.equation(block: true, alt: "open parenthesis m plus 6 close parenthesis open parenthesis m squared minus 7 m minus 30 close parenthesis")[$( m + 6 ) ( m^(2) − 7 m − 30 )$]

#math.equation(block: true, alt: "open parenthesis 4 y minus 1 close parenthesis open parenthesis 6 y squared minus 12 y plus 5 close parenthesis")[$( 4 y − 1 ) ( 6 y^(2) − 12 y + 5 )$]

#solutionbox[
#math.equation(block: true, alt: "24 y cubed minus 54 y squared plus 32 y minus 5")[$24 y^(3) − 54 y^(2) + 32 y − 5$]
]

#strong[Multiply Special Products]

In the following exercises, square each binomial using the Binomial Squares Pattern.

#math.equation(block: true, alt: "open parenthesis 2 x minus y close parenthesis squared")[$attach(( 2 x − y ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis x plus the fraction 3 over 4 close parenthesis squared")[$attach(( x + frac(3, 4) ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "x squared plus the fraction 3 over 2 x plus the fraction 9 over 16")[$x^(2) + frac(3, 2) x + frac(9, 16)$]
]

#math.equation(block: true, alt: "open parenthesis 8 p cubed minus 3 close parenthesis squared")[$attach(( 8 p^(3) − 3 ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 5 p plus 7 q close parenthesis squared")[$attach(( 5 p + 7 q ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "25 p squared plus 70 p q plus 49 q squared")[$25 p^(2) + 70 p q + 49 q^(2)$]
]

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

#math.equation(block: true, alt: "open parenthesis 3 y plus 5 close parenthesis open parenthesis 3 y minus 5 close parenthesis")[$( 3 y + 5 ) ( 3 y − 5 )$]

#math.equation(block: true, alt: "open parenthesis 6 x plus y close parenthesis open parenthesis 6 x minus y close parenthesis")[$( 6 x + y ) ( 6 x − y )$]

#solutionbox[
#math.equation(block: true, alt: "36 x squared minus y squared")[$36 x^(2) − y^(2)$]
]

#math.equation(block: true, alt: "open parenthesis a plus the fraction 2 over 3 b close parenthesis open parenthesis a minus the fraction 2 over 3 b close parenthesis")[$( a + frac(2, 3) b ) ( a − frac(2, 3) b )$]

#math.equation(block: true, alt: "open parenthesis 12 x cubed minus 7 y squared close parenthesis open parenthesis 12 x cubed plus 7 y squared close parenthesis")[$( 12 x^(3) − 7 y^(2) ) ( 12 x^(3) + 7 y^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "144 x to the power 6 minus 49 y to the power 4")[$144 x^(6) − 49 y^(4)$]
]

#math.equation(block: true, alt: "open parenthesis 13 a squared minus 8 b to the power 4 close parenthesis open parenthesis 13 a squared plus 8 b to the power 4 close parenthesis")[$( 13 a^(2) − 8 b^(4) ) ( 13 a^(2) + 8 b^(4) )$]


==== Divide Monomials

#strong[Divide Monomials]

In the following exercises, divide the monomials.

#math.equation(block: true, alt: "72 p to the power 12 divided by 8 p cubed")[$72 p^(12) ÷ 8 p^(3)$]

#solutionbox[
#math.equation(block: true, alt: "9 p to the power 9")[$9 p^(9)$]
]

#math.equation(block: true, alt: "−26 a to the power 8 divided by open parenthesis 2 a squared close parenthesis")[$−26 a^(8) ÷ ( 2 a^(2) )$]

#math.equation(block: true, alt: "the fraction 45 y to the power 6 over −15 y to the power 10")[$frac(45 y^(6), −15 y^(10))$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 over y to the power 4")[$− frac(3, y^(4))$]
]

#math.equation(block: true, alt: "the fraction −30 x to the power 8 over −36 x to the power 9")[$frac(−30 x^(8), −36 x^(9))$]

#math.equation(block: true, alt: "the fraction 28 a to the power 9 b over 7 a to the power 4 b cubed")[$frac(28 a^(9) b, 7 a^(4) b^(3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 a to the power 5 over b squared")[$frac(4 a^(5), b^(2))$]
]

#math.equation(block: true, alt: "the fraction 11 u to the power 6 v cubed over 55 u squared v to the power 8")[$frac(11 u^(6) v^(3), 55 u^(2) v^(8))$]

#math.equation(block: true, alt: "the fraction open parenthesis 5 m to the power 9 n cubed close parenthesis open parenthesis 8 m cubed n squared close parenthesis over open parenthesis 10 m n to the power 4 close parenthesis open parenthesis m squared n to the power 5 close parenthesis")[$frac(( 5 m^(9) n^(3) ) ( 8 m^(3) n^(2) ), ( 10 m n^(4) ) ( m^(2) n^(5) ))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 m to the power 9 over n to the power 4")[$frac(4 m^(9), n^(4))$]
]

#math.equation(block: true, alt: "the fraction open parenthesis 42 r squared s to the power 4 close parenthesis open parenthesis 54 r s squared close parenthesis over open parenthesis 6 r s cubed close parenthesis open parenthesis 9 s close parenthesis")[$frac(( 42 r^(2) s^(4) ) ( 54 r s^(2) ), ( 6 r s^(3) ) ( 9 s ))$]

#strong[Divide a Polynomial by a Monomial]

In the following exercises, divide each polynomial by the monomial

#math.equation(block: true, alt: "open parenthesis 54 y to the power 4 minus 24 y cubed close parenthesis divided by open parenthesis −6 y squared close parenthesis")[$( 54 y^(4) − 24 y^(3) ) ÷ ( −6 y^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "−9 y squared plus 4 y")[$−9 y^(2) + 4 y$]
]

#math.equation(block: true, alt: "the fraction 63 x cubed y squared minus 99 x squared y cubed minus 45 x to the power 4 y cubed over 9 x squared y squared")[$frac(63 x^(3) y^(2) − 99 x^(2) y^(3) − 45 x^(4) y^(3), 9 x^(2) y^(2))$]

#math.equation(block: true, alt: "the fraction 12 x squared plus 4 x minus 3 over −4 x")[$frac(12 x^(2) + 4 x − 3, −4 x)$]

#solutionbox[
#math.equation(block: true, alt: "−3 x minus 1 plus the fraction 3 over 4 x")[$−3 x − 1 + frac(3, 4 x)$]
]

#strong[Divide Polynomials using Long Division]

In the following exercises, divide each polynomial by the binomial.

#math.equation(block: true, alt: "open parenthesis 4 x squared minus 21 x minus 18 close parenthesis divided by open parenthesis x minus 6 close parenthesis")[$( 4 x^(2) − 21 x − 18 ) ÷ ( x − 6 )$]

#math.equation(block: true, alt: "open parenthesis y squared plus 2 y plus 18 close parenthesis divided by open parenthesis y plus 5 close parenthesis")[$( y^(2) + 2 y + 18 ) ÷ ( y + 5 )$]

#solutionbox[
#math.equation(block: true, alt: "y minus 3 plus the fraction 33 over y plus 5")[$y − 3 + frac(33, y + 5)$]
]

#math.equation(block: true, alt: "open parenthesis n cubed minus 2 n squared minus 6 n plus 27 close parenthesis divided by open parenthesis n plus 3 close parenthesis")[$( n^(3) − 2 n^(2) − 6 n + 27 ) ÷ ( n + 3 )$]

#math.equation(block: true, alt: "open parenthesis a cubed minus 1 close parenthesis divided by open parenthesis a plus 1 close parenthesis")[$( a^(3) − 1 ) ÷ ( a + 1 )$]

#solutionbox[
#math.equation(block: true, alt: "a squared minus a plus 1 minus the fraction 2 over a plus 1")[$a^(2) − a + 1 − frac(2, a + 1)$]
]

#strong[Divide Polynomials using Synthetic Division]

#emph[In the following exercises], use synthetic Division to find the quotient and remainder.

#math.equation(block: false, alt: "x cubed minus 3 x squared minus 4 x plus 12")[$x^(3) − 3 x^(2) − 4 x + 12$] is divided by #math.equation(block: false, alt: "x plus 2")[$x + 2$]

#math.equation(block: false, alt: "2 x cubed minus 11 x squared plus 11 x plus 12")[$2 x^(3) − 11 x^(2) + 11 x + 12$] is divided by #math.equation(block: false, alt: "x minus 3")[$x − 3$]

#solutionbox[
#math.equation(block: true, alt: "2 x squared minus 5 x minus 4 ; 0")[$2 x^(2) − 5 x − 4 ";" #h(0.2em) 0$]
]

#math.equation(block: false, alt: "x to the power 4 plus x squared plus 6 x minus 10")[$x^(4) + x^(2) + 6 x − 10$] is divided by #math.equation(block: false, alt: "x plus 2")[$x + 2$]

#strong[Divide Polynomial Functions]

In the following exercises, divide.

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 15 x plus 54")[$f ( x ) = x^(2) − 15 x + 54$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x minus 9 ,")[$g ( x ) = x − 9 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −2 close parenthesis")[$( frac(f, g) ) ( −2 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals x minus 6")[$( frac(f, g) ) ( x ) = x − 6$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis −2 close parenthesis equals −8")[$( frac(f, g) ) ( −2 ) = −8$]
]

For functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus x squared minus 7 x plus 2")[$f ( x ) = x^(3) + x^(2) − 7 x + 2$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x minus 2 ,")[$g ( x ) = x − 2 ,$] find ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 3 close parenthesis")[$( frac(f, g) ) ( 3 )$]

#strong[Use the Remainder and Factor Theorem]

In the following exercises, use the Remainder Theorem to find the remainder.

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 4 x minus 9")[$f ( x ) = x^(3) − 4 x − 9$] is divided by #math.equation(block: false, alt: "x plus 2")[$x + 2$]

#solutionbox[
#math.equation(block: true, alt: "−9")[$−9$]
]

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x cubed minus 6 x minus 24")[$f ( x ) = 2 x^(3) − 6 x − 24$] divided by #math.equation(block: false, alt: "x minus 3")[$x − 3$]

In the following exercises, use the Factor Theorem to determine if #math.equation(block: false, alt: "x minus c")[$x − c$] is a factor of the polynomial function.

Determine whether #math.equation(block: false, alt: "x minus 2")[$x − 2$] is a factor of #math.equation(block: false, alt: "x cubed minus 7 x squared plus 7 x minus 6")[$x^(3) − 7 x^(2) + 7 x − 6$].

#solutionbox[
no
]

Determine whether #math.equation(block: false, alt: "x minus 3")[$x − 3$] is a factor of #math.equation(block: false, alt: "x cubed minus 7 x squared plus 11 x plus 3")[$x^(3) − 7 x^(2) + 11 x + 3$].


=== Chapter Practice Test

For the polynomial #math.equation(block: false, alt: "8 y to the power 4 minus 3 y squared plus 1")[$8 y^(4) − 3 y^(2) + 1$]

ⓐ Is it a monomial, binomial, or trinomial? ⓑ What is its degree?

#solutionbox[
ⓐ trinomial ⓑ 4
]

#math.equation(block: true, alt: "open parenthesis 5 a squared plus 2 a minus 12 close parenthesis open parenthesis 9 a squared plus 8 a minus 4 close parenthesis")[$( 5 a^(2) + 2 a − 12 ) ( 9 a^(2) + 8 a − 4 )$]

#math.equation(block: true, alt: "open parenthesis 10 x squared minus 3 x plus 5 close parenthesis minus open parenthesis 4 x squared minus 6 close parenthesis")[$( 10 x^(2) − 3 x + 5 ) − ( 4 x^(2) − 6 )$]

#solutionbox[
#math.equation(block: true, alt: "6 x squared minus 3 x plus 11")[$6 x^(2) − 3 x + 11$]
]

#math.equation(block: true, alt: "open parenthesis minus the fraction 3 over 4 close parenthesis cubed")[$attach(( − frac(3, 4) ), t: 3)$]

#math.equation(block: true, alt: "x to the power −3 x to the power 4")[$x^(−3) x^(4)$]

#solutionbox[
#math.equation(block: true, alt: "x")[$x$]
]

#math.equation(block: true, alt: "the fraction 5 to the power 6 over 5 to the power 8")[$frac(5^(6), 5^(8))$]

#math.equation(block: true, alt: "open parenthesis 47 a to the power 18 b to the power 23 c to the power 5 close parenthesis to the power 0")[$attach(( 47 a^(18) b^(23) c^(5) ), t: 0)$]

#solutionbox[
#math.equation(block: true, alt: "1")[$1$]
]

#math.equation(block: true, alt: "4 to the power −1")[$4^(−1)$]

#math.equation(block: true, alt: "open parenthesis 2 y close parenthesis to the power −3")[$attach(( 2 y ), t: −3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 8 y cubed")[$frac(1, 8 y^(3))$]
]

#math.equation(block: true, alt: "p to the power −3 times p to the power −8")[$p^(−3) · p^(−8)$]

#math.equation(block: true, alt: "the fraction x to the power 4 over x to the power −5")[$frac(x^(4), x^(−5))$]

#solutionbox[
#math.equation(block: true, alt: "x to the power 9")[$x^(9)$]
]

#math.equation(block: true, alt: "open parenthesis 3 x to the power −3 close parenthesis squared")[$attach(( 3 x^(−3) ), t: 2)$]

#math.equation(block: true, alt: "the fraction 24 r cubed s over 6 r squared s to the power 7")[$frac(24 r^(3) s, 6 r^(2) s^(7))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 r over s to the power 6")[$frac(4 r, s^(6))$]
]

#math.equation(block: true, alt: "open parenthesis the fraction x to the power 4 y to the power 9 over x to the power −3 close parenthesis squared")[$attach(( frac(x^(4) y^(9), x^(−3)) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 8 x y cubed close parenthesis open parenthesis −6 x to the power 4 y to the power 6 close parenthesis")[$( 8 x y^(3) ) ( −6 x^(4) y^(6) )$]

#solutionbox[
#math.equation(block: true, alt: "−48 x to the power 5 y to the power 9")[$−48 x^(5) y^(9)$]
]

#math.equation(block: true, alt: "4 u open parenthesis u squared minus 9 u plus 1 close parenthesis")[$4 u ( u^(2) − 9 u + 1 )$]

#math.equation(block: true, alt: "open parenthesis m plus 3 close parenthesis open parenthesis 7 m minus 2 close parenthesis")[$( m + 3 ) ( 7 m − 2 )$]

#solutionbox[
#math.equation(block: true, alt: "7 m squared plus 19 m minus 6")[$7 m^(2) + 19 m − 6$]
]

#math.equation(block: true, alt: "open parenthesis n minus 8 close parenthesis open parenthesis n squared minus 4 n plus 11 close parenthesis")[$( n − 8 ) ( n^(2) − 4 n + 11 )$]

#math.equation(block: true, alt: "open parenthesis 4 x minus 3 close parenthesis squared")[$attach(( 4 x − 3 ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "16 x squared minus 24 x plus 9")[$16 x^(2) − 24 x + 9$]
]

#math.equation(block: true, alt: "open parenthesis 5 x plus 2 y close parenthesis open parenthesis 5 x minus 2 y close parenthesis")[$( 5 x + 2 y ) ( 5 x − 2 y )$]

#math.equation(block: true, alt: "open parenthesis 15 x y cubed minus 35 x squared y close parenthesis divided by 5 x y")[$( 15 x y^(3) − 35 x^(2) y ) ÷ 5 x y$]

#solutionbox[
#math.equation(block: true, alt: "3 y squared minus 7 x")[$3 y^(2) − 7 x$]
]

#math.equation(block: true, alt: "open parenthesis 3 x cubed minus 10 x squared plus 7 x plus 10 close parenthesis divided by open parenthesis 3 x plus 2 close parenthesis")[$( 3 x^(3) − 10 x^(2) + 7 x + 10 ) ÷ ( 3 x + 2 )$]

Use the Factor Theorem to determine if #math.equation(block: false, alt: "x plus 3")[$x + 3$] a factor of #math.equation(block: false, alt: "x cubed plus 8 x squared plus 21 x plus 18 .")[$x^(3) + 8 x^(2) + 21 x + 18 .$]

#solutionbox[
yes
]

ⓐ Convert 112,000 to scientific notation. ⓑ Convert #math.equation(block: false, alt: "5.25 times 10 to the power −4")[$5.25 #h(0.2em) × #h(0.2em) 10^(−4)$] to decimal form.

In the following exercises, simplify and write your answer in exponential notation.

#math.equation(block: true, alt: "open parenthesis 2.4 times 10 to the power 8 close parenthesis open parenthesis 2 times 10 to the power −5 close parenthesis")[$( 2.4 #h(0.2em) × #h(0.2em) 10^(8) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−5) )$]

#solutionbox[
#math.equation(block: true, alt: "4.8 times 10 cubed")[$4.8 #h(0.2em) × #h(0.2em) 10^(3)$]
]

#math.equation(block: true, alt: "the fraction 9 times 10 to the power 4 over 3 times 10 to the power −1")[$frac(9 #h(0.2em) × #h(0.2em) 10^(4), 3 #h(0.2em) × #h(0.2em) 10^(−1))$]

For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 6 x squared minus 3 x minus 9")[$f ( x ) = 6 x^(2) − 3 x − 9$] find: #linebreak() ⓐ #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis")[$f ( 3 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis")[$f ( −2 )$] ⓒ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis")[$f ( 0 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "36")[$36$] ⓑ #math.equation(block: false, alt: "21")[$21$] ⓒ #math.equation(block: false, alt: "minus 9")[$− 9$]
]

For #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x squared minus 3 x minus 5")[$f ( x ) = 2 x^(2) − 3 x − 5$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 3 x squared minus 4 x plus 1 ,")[$g ( x ) = 3 x^(2) − 4 x + 1 ,$] find #linebreak() ⓐ #math.equation(block: false, alt: "open parenthesis f plus g close parenthesis open parenthesis x close parenthesis")[$( f + g ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis f plus g close parenthesis open parenthesis 1 close parenthesis")[$( f + g ) ( 1 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis f minus g close parenthesis open parenthesis x close parenthesis")[$( f − g ) ( x )$] ⓓ #math.equation(block: false, alt: "open parenthesis f minus g close parenthesis open parenthesis −2 close parenthesis")[$( f − g ) ( −2 )$]

For functions #linebreak() #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 x squared minus 23 x minus 36")[$f ( x ) = 3 x^(2) − 23 x − 36$] and #linebreak() #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x minus 9 ,")[$g ( x ) = x − 9 ,$] find #linebreak() ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis")[$( frac(f, g) ) ( x )$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 3 close parenthesis")[$( frac(f, g) ) ( 3 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis x close parenthesis equals 3 x plus 4")[$( frac(f, g) ) ( x ) = 3 x + 4$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction f over g close parenthesis open parenthesis 3 close parenthesis equals 13")[$( frac(f, g) ) ( 3 ) = 13$]
]

A hiker drops a pebble from a bridge 240 feet above a canyon. The function #math.equation(block: false, alt: "h open parenthesis t close parenthesis equals −16 t squared plus 240")[$h ( t ) = −16 t^(2) + 240$] gives the height of the pebble #math.equation(block: false, alt: "t")[$t$] seconds after it was dropped. Find the height when #math.equation(block: false, alt: "t equals 3 .")[$t = 3 .$]