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== 10.6#h(0.6em)Introduction to Factoring Polynomials

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

Factor #math.equation(block: false, alt: "56")[$56$] into primes. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 times 2 times 2 times 7")[$2 ⋅ 2 ⋅ 2 ⋅ 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "−3 open parenthesis 6 a plus 11 close parenthesis .")[$−3 ( 6 a + 11 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 18 a minus 33")[$− 18 a − 33$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "4 x squared open parenthesis x squared plus 3 x minus 1 close parenthesis .")[$4 x^(2) ( x^(2) + 3 x − 1 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "4 x to the power 4 plus 12 x cubed minus 4 x squared")[$4 x^(4) + 12 x^(3) − 4 x^(2)$]
]
]


=== Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

#figure(figph[On the left, the equation 8 times 7 equals 56 is shown. 8 and 7 are labeled factors, 56 is labeled product. On the right, the equation 2x times parentheses x plus 3 equals 2 x squared plus 6x is shown. 2x and x plus 3 are labeled factors, 2 x squared plus 6x is labeled product. There is an arrow on top pointing to the right that says “multiply” in red. There is an arrow on the bottom pointing to the left that says “factor” in red.], alt: "On the left, the equation 8 times 7 equals 56 is shown. 8 and 7 are labeled factors, 56 is labeled product. On the right, the equation 2x times parentheses x plus 3 equals 2 x squared plus 6x is shown. 2x and x plus 3 are labeled factors, 2 x squared plus 6x is labeled product. There is an arrow on top pointing to the right that says “multiply” in red. There is an arrow on the bottom pointing to the left that says “factor” in red.", caption: none)

In The Language of Algebra we factored numbers to find the #strong[least common multiple] (LCM) of two or more numbers. Now we will factor expressions and find the #emph[greatest common factor] of two or more expressions. The method we use is similar to what we used to find the LCM.

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

The #strong[greatest common factor] (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
]

First we will find the greatest common factor of two numbers.

#examplebox("Example 1")[][
Find the greatest common factor of #math.equation(block: false, alt: "24")[$24$] and #math.equation(block: false, alt: "36 .")[$36 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[Step 1:] Factor each coefficient into primes. Write all variables with exponents in expanded form.], [Factor 24 and 36.], [#figure(figph[Factor trees demonstrating the prime factorization of 24 and 36.], alt: "Factor trees demonstrating the prime factorization of 24 and 36.", caption: none)]),
  [#strong[Step 2:] List all factors--matching common factors in a column.], [], [#figure(figph[Two lines of mathematical equations demonstrating the prime factorization of 24 and 36. 24 is shown as 2 x 2 x 2 x 3, and 36 as 2 x 2 x 3 x 3, with an underline beneath the latter.], alt: "Two lines of mathematical equations demonstrating the prime factorization of 24 and 36. 24 is shown as 2 x 2 x 2 x 3, and 36 as 2 x 2 x 3 x 3, with an underline beneath the latter.", caption: none)],
  [In each column, circle the common factors.], [Circle the 2, 2, and 3 that are shared by both numbers.], [#figure(figph[A step-by-step example of finding the Greatest Common Factor (GCF) of 24 and 36 using prime factorization, illustrating how common prime factors (2, 2, and 3) are multiplied to get the GCF, which is 12.], alt: "A step-by-step example of finding the Greatest Common Factor (GCF) of 24 and 36 using prime factorization, illustrating how common prime factors (2, 2, and 3) are multiplied to get the GCF, which is 12.", caption: none)],
  [#strong[Step 3:] Bring down the common factors that all expressions share.], [Bring down the 2, 2, 3 and then multiply.], [],
  [#strong[Step 4:] Multiply the factors.], [], [The GCF of 24 and 36 is 12.],
))

Notice that since the GCF is a factor of both numbers, #math.equation(block: false, alt: "24")[$24$] and #math.equation(block: false, alt: "36")[$36$] can be written as multiples of #math.equation(block: false, alt: "12 .")[$12 .$]

#math.equation(block: true, alt: "24 equals 12 times 2; 36 equals 12 times 3")[$24 = 12 · 2 \ 36 = 12 · 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "54 , 36 .")[$54 , 36 .$]

#solutionbox[
18
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "48 , 80 .")[$48 , 80 .$]

#solutionbox[
16
]
]

In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the #strong[greatest common factor].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Find the greatest common factor.]

+ Factor each coefficient into primes. Write all variables with exponents in expanded form.
+ List all factors—matching common factors in a column. In each column, circle the common factors.
+ Bring down the common factors that all expressions share.
+ Multiply the factors.
]

#examplebox("Example 2")[][
Find the greatest common factor of #math.equation(block: false, alt: "5 x and 15 .")[$5 x #h(0.2em) "and" #h(0.2em) 15 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Factor each number into primes. #linebreak() 
  Circle the common factors in each column. #linebreak() 
  Bring down the common factors.], [#figure(figph[An image shows the calculation for the Greatest Common Factor (GCF) of 5x and 15. It factors 5x as 5\*x and 15 as 3\*5. The common factor '5' is circled, resulting in GCF = 5.], alt: "An image shows the calculation for the Greatest Common Factor (GCF) of 5x and 15. It factors 5x as 5*x and 15 as 3*5. The common factor '5' is circled, resulting in GCF = 5.", caption: none)]),
  [], [The GCF of 5x and 15 is 5.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "7 y , 14 .")[$7 y , #h(0.2em) 14 .$]

#solutionbox[
7
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "22 , 11 m .")[$22 , #h(0.2em) 11 m .$]

#solutionbox[
11
]
]

In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

#examplebox("Example 3")[][
Find the greatest common factor of #math.equation(block: false, alt: "12 x squared")[$12 x^(2)$] and #math.equation(block: false, alt: "18 x cubed .")[$18 x^(3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Factor each coefficient into primes and write
 #linebreak() the variables with exponents in expanded form.
  #linebreak() Circle the common factors in each column.
 #linebreak() Bring down the common factors.
 #linebreak() Multiply the factors.], [#figure(figph[An algebraic problem illustrating the calculation of the Greatest Common Factor (GCF) for 12x^2 and 18x^3. The image factors both expressions, using pink circles to highlight common factors that combine to form the GCF, 6x^2.], alt: "An algebraic problem illustrating the calculation of the Greatest Common Factor (GCF) for 12x^2 and 18x^3. The image factors both expressions, using pink circles to highlight common factors that combine to form the GCF, 6x^2.", caption: none)]),
  [], [#math.equation(block: false, alt: "The GCF of 12 x squared and 18 x cubed is 6 x squared")[$"The GCF of" #h(0.2em) 12 x^(2) #h(0.2em) "and" #h(0.2em) 18 x^(3) #h(0.2em) "is" #h(0.2em) 6 x^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "16 x squared , 24 x cubed .")[$16 x^(2) , #h(0.2em) 24 x^(3) .$]

#solutionbox[
8#emph[x]#super[2]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "27 y cubed , 18 y to the power 4 .")[$27 y^(3) , #h(0.2em) 18 y^(4) .$]

#solutionbox[
9#emph[y]#super[3]
]
]

#examplebox("Example 4")[][
Find the greatest common factor of #math.equation(block: false, alt: "14 x cubed , 8 x squared , 10 x .")[$14 x^(3) , #h(0.2em) 8 x^(2) , #h(0.2em) 10 x .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Factor each coefficient into primes and write
 #linebreak() the variables with exponents in expanded form.
 #linebreak() Circle the common factors in each column.
 #linebreak() Bring down the common factors.
 #linebreak() Multiply the factors.], [#figure(figph[An image illustrating the process of finding the Greatest Common Factor (GCF) of 14x^3, 8x^2, and 10x. The prime factorization of each term is listed, with common factors (2 and x) circled, leading to a GCF of 2x.], alt: "An image illustrating the process of finding the Greatest Common Factor (GCF) of 14x^3, 8x^2, and 10x. The prime factorization of each term is listed, with common factors (2 and x) circled, leading to a GCF of 2x.", caption: none)]),
  [], [#math.equation(block: false, alt: "The GCF of 14 x cubed and 8 x squared , and 10 x is 2 x")[$"The GCF of" #h(0.2em) 14 x^(3) #h(0.2em) "and" #h(0.2em) 8 x^(2) ", and" 10 x #h(0.2em) "is" #h(0.2em) 2 x$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "21 x cubed , 9 x squared , 15 x .")[$21 x^(3) , #h(0.2em) 9 x^(2) , #h(0.2em) 15 x .$]

#solutionbox[
3#emph[x]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "25 m to the power 4 , 35 m cubed , 20 m squared .")[$25 m^(4) , #h(0.2em) 35 m^(3) , #h(0.2em) 20 m^(2) .$]

#solutionbox[
5#emph[m]#super[2]
]
]


=== Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, #math.equation(block: false, alt: "12")[$12$] as #math.equation(block: false, alt: "2 times 6 or 3 times 4 ),")[$2 · 6 #h(0.2em) "or" #h(0.2em) 3 · 4 "),"$] in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

#math.equation(block: true, alt: "2 open parenthesis x, plus, 7 close parenthesis factors; 2 times x, plus, 2 times 7; 2 x, plus, 14 product")[$2 ( x & + & 7 ) #h(0.2em) "factors" \ 2 · x & + & 2 · 7 \ 2 x & + & 14 #h(0.2em) "product"$]Here, we will start with a product, like #math.equation(block: false, alt: "2 x plus 14 ,")[$2 x + 14 ,$] and end with its factors, #math.equation(block: false, alt: "2 open parenthesis x plus 7 close parenthesis .")[$2 ( x + 7 ) .$] To do this we apply the #strong[Distributive Property] “in reverse”.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Distributive Property]

If #math.equation(block: false, alt: "a , b , c")[$a , #h(0.2em) b , #h(0.2em) c$] are real numbers, then

#math.equation(block: true, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c and a b plus a c equals a open parenthesis b plus c close parenthesis")[$a ( b + c ) = a b + a c #h(2em) "and" #h(2em) a b + a c = a ( b + c )$]
]

The form on the left is used to multiply. The form on the right is used to factor.

So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

#examplebox("Example 5")[][
Factor: #math.equation(block: false, alt: "2 x plus 14 .")[$2 x + 14 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[Step 1:] Find the GCF of all the terms of the polynomial.], [Find the GCF of 2x and 14.], [#figure(figph[A math problem demonstrating how to find the Greatest Common Factor (GCF) of 2x and 14. The number 2 is circled as the common factor, yielding GCF = 2.], alt: "A math problem demonstrating how to find the Greatest Common Factor (GCF) of 2x and 14. The number 2 is circled as the common factor, yielding GCF = 2.", caption: none)]),
  [#strong[Step 2:] Rewrite each term as a product using the GCF.], [Rewrite 2x and 14 as products of their GCF, 2. #linebreak() #math.equation(block: false, alt: "2 x equals 2 times x")[$2 x = 2 ⋅ x$] #linebreak() #math.equation(block: false, alt: "14 equals 2 times 7")[$14 = 2 ⋅ 7$]], [#figure(figph[A mathematical expression showing the process of factoring out a common number. The first line is 2x + 14, and the second line breaks down 14 into 2 \* 7, highlighting the common factor of 2 in red as 2 \* x + 2 \* 7.], alt: "A mathematical expression showing the process of factoring out a common number. The first line is 2x + 14, and the second line breaks down 14 into 2 * 7, highlighting the common factor of 2 in red as 2 * x + 2 * 7.", caption: none)],
  [#strong[Step 3:] Use the Distributive Property 'in reverse' to factor the expression.], [], [#math.equation(block: false, alt: "2 open parenthesis x plus 7 close parenthesis")[$2 ( x + 7 )$]],
  [#strong[Step 4:] Check by multiplying the factors.], [], [Check: #linebreak() #figure(figph[An image illustrating the distributive property in algebra, showing the expansion of 2(x+7) into 2x+14. The steps demonstrate multiplying 2 by both x and 7, resulting in 2x + 14, confirmed with a checkmark.], alt: "An image illustrating the distributive property in algebra, showing the expansion of 2(x+7) into 2x+14. The steps demonstrate multiplying 2 by both x and 7, resulting in 2x + 14, confirmed with a checkmark.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "4 x plus 12 .")[$4 x + 12 .$]

#solutionbox[
4(#emph[x] + 3)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "6 a plus 24 .")[$6 a + 24 .$]

#solutionbox[
6(#emph[a] + 4)
]
]

Notice that in , we used the word #emph[factor] as both a noun and a verb:

#math.equation(block: true, alt: "Noun, 7 is a factor of 14; Verb, factor 2 from 2 x plus 14")[$"Noun" &  &  & 7 #h(0.2em) "is a factor of" #h(0.2em) 14 \ "Verb" &  &  & "factor" #h(0.2em) 2 #h(0.2em) "from" #h(0.2em) 2 x + 14$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor the greatest common factor from a polynomial.]

+ Find the GCF of all the terms of the polynomial.
+ Rewrite each term as a product using the GCF.
+ Use the Distributive Property ‘in reverse’ to factor the expression.
+ Check by multiplying the factors.
]

#examplebox("Example 6")[][
Factor: #math.equation(block: false, alt: "3 a plus 3 .")[$3 a + 3 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#figure(figph[A math problem illustrating how to find the Greatest Common Factor (GCF) of 3a and 3. The solution shows 3a factored as 3 \* a and 3 as 3, highlighting 3 as the common factor, resulting in GCF = 3.], alt: "A math problem illustrating how to find the Greatest Common Factor (GCF) of 3a and 3. The solution shows 3a factored as 3 * a and 3 as 3, highlighting 3 as the common factor, resulting in GCF = 3.", caption: none)], []),
  [], [#figure(figph[The mathematical expression '3a + 3' is displayed in a bold, dark gray font against a plain white background.], alt: "The mathematical expression '3a + 3' is displayed in a bold, dark gray font against a plain white background.", caption: none)],
  [Rewrite each term as a product using the GCF.], [#figure(figph[A mathematical expression showing 3 multiplied by 'a' plus 3 multiplied by 1, which can be factored as 3(a+1).], alt: "A mathematical expression showing 3 multiplied by 'a' plus 3 multiplied by 1, which can be factored as 3(a+1).", caption: none)],
  [Use the Distributive Property 'in reverse' to factor the GCF.], [#figure(figph[A mathematical expression: 3(a + 1).], alt: "A mathematical expression: 3(a + 1).", caption: none)],
  [Check by multiplying the factors to get the original polynomial.], [],
  [#figure(figph[Illustration of the distributive property, showing 3(a+1) expanding to 3a + 3 with a checkmark for correctness.], alt: "Illustration of the distributive property, showing 3(a+1) expanding to 3a + 3 with a checkmark for correctness.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "9 a plus 9 .")[$9 a + 9 .$]

#solutionbox[
9(#emph[a] + 1)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "11 x plus 11 .")[$11 x + 11 .$]

#solutionbox[
11(#emph[x] + 1)
]
]

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

#examplebox("Example 7")[][
Factor: #math.equation(block: false, alt: "12 x minus 60 .")[$12 x − 60 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#figure(figph[Steps to find the Greatest Common Factor (GCF) of 12x and 60 using prime factorization, showing the common factors circled and the final GCF as 12.], alt: "Steps to find the Greatest Common Factor (GCF) of 12x and 60 using prime factorization, showing the common factors circled and the final GCF as 12.", caption: none)], []),
  [], [#figure(figph[The mathematical expression '12x - 60' is displayed in a dark gray font on a white background.], alt: "The mathematical expression '12x - 60' is displayed in a dark gray font on a white background.", caption: none)],
  [Rewrite each term as a product using the GCF.], [#figure(figph[The mathematical expression 12 multiplied by x minus 12 multiplied by 5 is shown.], alt: "The mathematical expression 12 multiplied by x minus 12 multiplied by 5 is shown.", caption: none)],
  [Factor the GCF.], [#figure(figph[A mathematical expression shows twelve multiplied by the quantity x minus five, written as 12(x - 5).], alt: "A mathematical expression shows twelve multiplied by the quantity x minus five, written as 12(x - 5).", caption: none)],
  [Check by multiplying the factors.], [],
  [#figure(figph[Applying the distributive property to simplify the algebraic expression 12(x-5) step-by-step, resulting in 12x-60.], alt: "Applying the distributive property to simplify the algebraic expression 12(x-5) step-by-step, resulting in 12x-60.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "11 x minus 44 .")[$11 x − 44 .$]

#solutionbox[
11(#emph[x] − 4)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "13 y minus 52 .")[$13 y − 52 .$]

#solutionbox[
13(#emph[y] − 4)
]
]

Now we’ll factor the #strong[greatest common factor] from a #strong[trinomial]. We start by finding the GCF of all three terms.

#examplebox("Example 8")[][
Factor: #math.equation(block: false, alt: "3 y squared plus 6 y plus 9 .")[$3 y^(2) + 6 y + 9 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#figure(figph[Finding the GCF of 3y^2, 6y, and 9 using prime factorization. Each term's factors are listed, and the common factor, 3, is circled, showing GCF = 3.], alt: "Finding the GCF of 3y^2, 6y, and 9 using prime factorization. Each term's factors are listed, and the common factor, 3, is circled, showing GCF = 3.", caption: none)], []),
  [], [#figure(figph[A mathematical expression displaying 3y squared plus 6y plus 9.], alt: "A mathematical expression displaying 3y squared plus 6y plus 9.", caption: none)],
  [Rewrite each term as a product using the GCF.], [#figure(figph[A mathematical expression displaying 3 multiplied by y squared, plus 3 multiplied by 2y, plus 3 multiplied by 3. The number 3 is highlighted in red in each term of the expression.], alt: "A mathematical expression displaying 3 multiplied by y squared, plus 3 multiplied by 2y, plus 3 multiplied by 3. The number 3 is highlighted in red in each term of the expression.", caption: none)],
  [Factor the GCF.], [#figure(figph[A mathematical expression showing 3 multiplied by the quantity (y-squared plus 2y plus 3), as 3(y^2 + 2y + 3).], alt: "A mathematical expression showing 3 multiplied by the quantity (y-squared plus 2y plus 3), as 3(y^2 + 2y + 3).", caption: none)],
  [Check by multiplying.], [],
  [#figure(figph[An algebraic expression 3(y^2 + 2y + 3) is expanded using the distributive property, showing the steps to arrive at the simplified form 3y^2 + 6y + 9, which is marked as correct.], alt: "An algebraic expression 3(y^2 + 2y + 3) is expanded using the distributive property, showing the steps to arrive at the simplified form 3y^2 + 6y + 9, which is marked as correct.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "4 y squared plus 8 y plus 12 .")[$4 y^(2) + 8 y + 12 .$]

#solutionbox[
4(#emph[y]#super[2] + 2#emph[y] + 3)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "6 x squared plus 42 x minus 12 .")[$6 x^(2) + 42 x − 12 .$]

#solutionbox[
6(#emph[x]#super[2] + 7#emph[x] − 2)
]
]

In the next example, we factor a variable from a #strong[binomial].

#examplebox("Example 9")[][
Factor: #math.equation(block: false, alt: "6 x squared plus 5 x .")[$6 x^(2) + 5 x .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "6 x squared plus 5 x")[$6 x^(2) + 5 x$]]),
  [Find the GCF of #math.equation(block: false, alt: "6 x squared")[$6 x^(2)$] and #math.equation(block: false, alt: "5 x")[$5 x$] and the math that goes with it.], [#figure(figph[This image illustrates the process of finding the Greatest Common Factor (GCF) of 6x^2 and 5x by factoring. The common factor 'x' is highlighted, leading to a GCF of x.], alt: "This image illustrates the process of finding the Greatest Common Factor (GCF) of 6x^2 and 5x by factoring. The common factor 'x' is highlighted, leading to a GCF of x.", caption: none)],
  [Rewrite each term as a product.], [#figure(figph[A mathematical expression showing 'x times 6x plus x times 5'. The variable 'x' is highlighted in red for both instances, emphasizing its presence in the terms.], alt: "A mathematical expression showing 'x times 6x plus x times 5'. The variable 'x' is highlighted in red for both instances, emphasizing its presence in the terms.", caption: none)],
  [Factor the GCF.], [#math.equation(block: false, alt: "x open parenthesis 6 x plus 5 close parenthesis")[$x ( 6 x + 5 )$]],
  [Check by multiplying.], [],
  [#math.equation(block: false, alt: "x open parenthesis 6 x plus 5 close parenthesis")[$x ( 6 x + 5 )$] #linebreak() #math.equation(block: false, alt: "x times 6 x plus x times 5")[$x ⋅ 6 x + x ⋅ 5$] #linebreak() #math.equation(block: false, alt: "6 x squared plus 5 x ✓")[$6 x^(2) + 5 x ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "9 x squared plus 7 x .")[$9 x^(2) + 7 x .$]

#solutionbox[
#emph[x] (9#emph[x] + 7)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "5 a squared minus 12 a .")[$5 a^(2) − 12 a .$]

#solutionbox[
#emph[a] (5#emph[a] − 12)
]
]

When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

#examplebox("Example 10")[][
Factor: #math.equation(block: false, alt: "4 x cubed minus 20 x squared .")[$4 x^(3) − 20 x^(2) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#figure(figph[An algebraic example showing how to find the Greatest Common Factor (GCF) of 4x^3 and 20x^2. Prime factors are listed and common factors are circled in pink, resulting in a GCF of 4x^2.], alt: "An algebraic example showing how to find the Greatest Common Factor (GCF) of 4x^3 and 20x^2. Prime factors are listed and common factors are circled in pink, resulting in a GCF of 4x^2.", caption: none)], [], [],
  [], [], [#figure(figph[A mathematical expression reads 4x^3 - 20x^2 in black text on a white background, formatted with standard mathematical notation for exponents and subtraction.], alt: "A mathematical expression reads 4x^3 - 20x^2 in black text on a white background, formatted with standard mathematical notation for exponents and subtraction.", caption: none)],
  [Rewrite each term.], [], [#figure(figph[The mathematical expression '4x^2 \* x - 4x^2 \* 5' is displayed, showing a subtraction operation between two terms, both involving 4x^2 multiplied by another factor.], alt: "The mathematical expression '4x^2 * x - 4x^2 * 5' is displayed, showing a subtraction operation between two terms, both involving 4x^2 multiplied by another factor.", caption: none)],
  [Factor the GCF.], [], [#figure(figph[The image displays the algebraic expression 4x^2(x-5) in black text on a white background, representing a polynomial in factored form. The terms are clearly visible, indicating multiplication.], alt: "The image displays the algebraic expression 4x^2(x-5) in black text on a white background, representing a polynomial in factored form. The terms are clearly visible, indicating multiplication.", caption: none)],
  [Check.], [#figure(figph[Algebraic expansion of 4x^2(x-5) to 4x^3 - 20x^2, illustrating the distributive property applied correctly and step-by-step.], alt: "Algebraic expansion of 4x^2(x-5) to 4x^3 - 20x^2, illustrating the distributive property applied correctly and step-by-step.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "2 x cubed plus 12 x squared .")[$2 x^(3) + 12 x^(2) .$]

#solutionbox[
2#emph[x]#super[2] (#emph[x] + 6)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "6 y cubed minus 15 y squared .")[$6 y^(3) − 15 y^(2) .$]

#solutionbox[
3#emph[y]#super[2] (2#emph[y] − 5)
]
]

#examplebox("Example 11")[][
Factor: #math.equation(block: false, alt: "21 y squared plus 35 y .")[$21 y^(2) + 35 y .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Find the GCF of #math.equation(block: false, alt: "21 y squared")[$21 y^(2)$] and #math.equation(block: false, alt: "35 y")[$35 y$]], [#figure(figph[Finding the Greatest Common Factor (GCF) of 21y^2 and 35y. The image demonstrates the prime factorization of both terms, highlighting common factors (7 and y) to calculate the GCF, which is 7y.], alt: "Finding the Greatest Common Factor (GCF) of 21y^2 and 35y. The image demonstrates the prime factorization of both terms, highlighting common factors (7 and y) to calculate the GCF, which is 7y.", caption: none)], []),
  [], [], [#figure(figph[The image displays the mathematical expression '21y^2 + 35y' in black text against a white background.], alt: "The image displays the mathematical expression '21y^2 + 35y' in black text against a white background.", caption: none)],
  [Rewrite each term.], [], [#figure(figph[A mathematical expression displaying the equation 7y multiplied by 3y, added to 7y multiplied by 5, with '7y' highlighted in red.], alt: "A mathematical expression displaying the equation 7y multiplied by 3y, added to 7y multiplied by 5, with '7y' highlighted in red.", caption: none)],
  [Factor the GCF.], [], [#figure(figph[A mathematical expression showing the term 7y multiplied by the binomial (3y + 5).], alt: "A mathematical expression showing the term 7y multiplied by the binomial (3y + 5).", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "18 y squared plus 63 y .")[$18 y^(2) + 63 y .$]

#solutionbox[
9#emph[y] (2#emph[y] + 7)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "32 k squared plus 56 k .")[$32 k^(2) + 56 k .$]

#solutionbox[
8#emph[k] (4#emph[k] + 7)
]
]

#examplebox("Example 12")[][
Factor: #math.equation(block: false, alt: "14 x cubed plus 8 x squared minus 10 x .")[$14 x^(3) + 8 x^(2) − 10 x .$]

#solutionbox[
Previously, we found the GCF of #math.equation(block: false, alt: "14 x cubed , 8 x squared , and 10 x")[$14 x^(3) , #h(0.2em) 8 x^(2) , #h(0.2em) "and" #h(0.2em) 10 x$] to be #math.equation(block: false, alt: "2 x .")[$2 x .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "14 x cubed plus 8 x squared minus 10 x")[$14 x^(3) + 8 x^(2) − 10 x$]]),
  [Rewrite each term using the GCF, 2x.], [#figure(figph[A mathematical expression: 2x \* 7x^2 + 2x \* 4x - 2x \* 5, showing a common factor of '2x' highlighted in red across three terms for potential factoring.], alt: "A mathematical expression: 2x * 7x^2 + 2x * 4x - 2x * 5, showing a common factor of '2x' highlighted in red across three terms for potential factoring.", caption: none)],
  [Factor the GCF.], [#math.equation(block: false, alt: "2 x open parenthesis 7 x squared plus 4 x minus 5 close parenthesis")[$2 x ( 7 x^(2) + 4 x − 5 )$]],
  [#figure(figph[Algebraic check using the distributive property: 2x(7x^2 + 4x - 5) = 2x\*7x^2 + 2x\*4x - 2x\*5 = 14x^3 + 8x^2 - 10x, confirmed with a checkmark.], alt: "Algebraic check using the distributive property: 2x(7x^2 + 4x - 5) = 2x*7x^2 + 2x*4x - 2x*5 = 14x^3 + 8x^2 - 10x, confirmed with a checkmark.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "18 y cubed minus 6 y squared minus 24 y .")[$18 y^(3) − 6 y^(2) − 24 y .$]

#solutionbox[
6#emph[y] (3#emph[y]#super[2] − #emph[y] − 4)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "16 x cubed plus 8 x squared minus 12 x .")[$16 x^(3) + 8 x^(2) − 12 x .$]

#solutionbox[
4#emph[x] (4#emph[x]#super[2] + 2#emph[x] − 3)
]
]

When the #strong[leading coefficient], the coefficient of the first term, is negative, we factor the negative out as part of the GCF.

#examplebox("Example 13")[][
Factor: #math.equation(block: false, alt: "−9 y minus 27 .")[$−9 y − 27 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9#emph[y] and 27 is 9.], [#figure(figph[The image demonstrates finding the Greatest Common Factor (GCF) of 9y and 27. It shows the prime factorization of 9y as 3\*3\*y and 27 as 3\*3\*3, with common factors 3 and 3 circled. The GCF is calculated as 9.], alt: "The image demonstrates finding the Greatest Common Factor (GCF) of 9y and 27. It shows the prime factorization of 9y as 3*3*y and 27 as 3*3*3, with common factors 3 and 3 circled. The GCF is calculated as 9.", caption: none)],
  [Since the expression −9y−27 has a negative leading coefficient, we use −9 as the GCF.], [],
  [], [#math.equation(block: false, alt: "minus 9 y minus 27")[$− 9 y − 27$]],
  [Rewrite each term using the GCF.], [#figure(figph[The image shows the mathematical expression -9 \* y + (-9) \* 3, which demonstrates the distributive property in an algebraic context. The number -9 is a common factor in both terms of the expression.], alt: "The image shows the mathematical expression -9 * y + (-9) * 3, which demonstrates the distributive property in an algebraic context. The number -9 is a common factor in both terms of the expression.", caption: none)],
  [Factor the GCF.], [#math.equation(block: false, alt: "minus 9 open parenthesis y plus 3 close parenthesis")[$− 9 ( y + 3 )$]],
  [#figure(figph[A mathematical check demonstrates the distributive property, simplifying -9(y + 3) to -9y - 27, confirmed correct with a checkmark.], alt: "A mathematical check demonstrates the distributive property, simplifying -9(y + 3) to -9y - 27, confirmed correct with a checkmark.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−5 y minus 35 .")[$−5 y − 35 .$]

#solutionbox[
−5(#emph[y] + 7)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−16 z minus 56 .")[$−16 z − 56 .$]

#solutionbox[
−8(2#emph[z] + 7)
]
]

Pay close attention to the signs of the terms in the next example.

#examplebox("Example 14")[][
Factor: #math.equation(block: false, alt: "−4 a squared plus 16 a .")[$−4 a^(2) + 16 a .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [The leading coefficient is negative, so the GCF will be negative.], [],
  [], [#figure(figph[The Greatest Common Factor (GCF) of 4a^2 and 16a is determined by prime factorization. Common factors (2, 2, a) are circled, and their product yields the GCF, which is 4a.], alt: "The Greatest Common Factor (GCF) of 4a^2 and 16a is determined by prime factorization. Common factors (2, 2, a) are circled, and their product yields the GCF, which is 4a.", caption: none)],
  [Since the leading coefficient is negative, the GCF is negative, −4#emph[a].], [],
  [], [#math.equation(block: false, alt: "−4 a squared plus 16 a")[$−4 a^(2) + 16 a$]],
  [Rewrite each term.], [#figure(figph[-4a \* a - (-4a) \* 4], alt: "-4a * a - (-4a) * 4", caption: none)],
  [Factor the GCF.], [#math.equation(block: false, alt: "minus 4 a open parenthesis a minus 4 close parenthesis")[$− 4 a ( a − 4 )$]],
  [Check on your own by multiplying.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−7 a squared plus 21 a .")[$−7 a^(2) + 21 a .$]

#solutionbox[
−7#emph[a] (#emph[a] − 3)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−6 x squared plus x .")[$−6 x^(2) + x .$]

#solutionbox[
−#emph[x] (6#emph[x] − 1)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://www.openstax.org/l/24factorgcf")[Factor GCF]
- #link("https://www.openstax.org/l/24factorbinomi")[Factor a Binomial]
- #link("https://www.openstax.org/l/24identifygcf")[Identify GCF]
]


=== Key Concepts

- #emph[Find the greatest common factor.]
+ Factor each coefficient into primes. Write all variables with exponents in expanded form.
+ List all factors—matching common factors in a column. In each column, circle the common factors.
+ Bring down the common factors that all expressions share.
+ Multiply the factors.
- #emph[Distributive Property]
- If #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], #math.equation(block: false, alt: "c")[$c$] are real numbers, then #linebreak() 
#math.equation(block: false, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c")[$a ( b + c ) = a b + a c$] and #math.equation(block: false, alt: "a b plus a c equals a open parenthesis b plus c close parenthesis")[$a b + a c = a ( b + c )$]
- #emph[Factor the greatest common factor from a polynomial.]
+ Find the GCF of all the terms of the polynomial.
+ Rewrite each term as a product using the GCF.
+ Use the Distributive Property ‘in reverse’ to factor the expression.
+ Check by multiplying the factors.


=== Section Exercises


==== Practice Makes Perfect

#strong[Find the Greatest Common Factor of Two or More Expressions]

In the following exercises, find the greatest common factor.

#math.equation(block: true, alt: "40 , 56")[$40 , 56$]

#math.equation(block: true, alt: "45 , 75")[$45 , 75$]

#solutionbox[
15
]

#math.equation(block: true, alt: "72 , 162")[$72 , 162$]

#math.equation(block: true, alt: "150 , 275")[$150 , 275$]

#solutionbox[
25
]

#math.equation(block: true, alt: "3 x , 12")[$3 x , 12$]

#math.equation(block: true, alt: "4 y , 28")[$4 y , 28$]

#solutionbox[
4
]

#math.equation(block: true, alt: "10 a , 50")[$10 a , 50$]

#math.equation(block: true, alt: "5 b , 30")[$5 b , 30$]

#solutionbox[
5
]

#math.equation(block: true, alt: "16 y , 24 y squared")[$16 y , #h(0.2em) 24 y^(2)$]

#math.equation(block: true, alt: "9 x , 15 x squared")[$9 x , #h(0.2em) 15 x^(2)$]

#solutionbox[
3#emph[x]
]

#math.equation(block: true, alt: "18 m cubed , 36 m squared")[$18 m^(3) , #h(0.2em) 36 m^(2)$]

#math.equation(block: true, alt: "12 p to the power 4 , 48 p cubed")[$12 p^(4) , #h(0.2em) 48 p^(3)$]

#solutionbox[
12#emph[p]#super[3]
]

#math.equation(block: true, alt: "10 x , 25 x squared , 15 x cubed")[$10 x , #h(0.2em) 25 x^(2) , #h(0.2em) 15 x^(3)$]

#math.equation(block: true, alt: "18 a , 6 a squared , 22 a cubed")[$18 a , #h(0.2em) 6 a^(2) , #h(0.2em) 22 a^(3)$]

#solutionbox[
2#emph[a]
]

#math.equation(block: true, alt: "24 u , 6 u squared , 30 u cubed")[$24 u , #h(0.2em) 6 u^(2) , #h(0.2em) 30 u^(3)$]

#math.equation(block: true, alt: "40 y , 10 y squared , 90 y cubed")[$40 y , #h(0.2em) 10 y^(2) , #h(0.2em) 90 y^(3)$]

#solutionbox[
10#emph[y]
]

#math.equation(block: true, alt: "15 a to the power 4 , 9 a to the power 5 , 21 a to the power 6")[$15 a^(4) , #h(0.2em) 9 a^(5) , #h(0.2em) 21 a^(6)$]

#math.equation(block: true, alt: "35 x cubed , 10 x to the power 4 , 5 x to the power 5")[$35 x^(3) , #h(0.2em) 10 x^(4) , #h(0.2em) 5 x^(5)$]

#solutionbox[
5#emph[x]#super[3]
]

#math.equation(block: true, alt: "27 y squared , 45 y cubed , 9 y to the power 4")[$27 y^(2) , #h(0.2em) 45 y^(3) , #h(0.2em) 9 y^(4)$]

#math.equation(block: true, alt: "14 b squared , 35 b cubed , 63 b to the power 4")[$14 b^(2) , #h(0.2em) 35 b^(3) , #h(0.2em) 63 b^(4)$]

#solutionbox[
7#emph[b]#super[2]
]

#strong[Factor the Greatest Common Factor from a Polynomial]

In the following exercises, factor the greatest common factor from each polynomial.

#math.equation(block: true, alt: "2 x plus 8")[$2 x + 8$]

#math.equation(block: true, alt: "5 y plus 15")[$5 y + 15$]

#solutionbox[
5(#emph[y] + 3)
]

#math.equation(block: true, alt: "3 a minus 24")[$3 a − 24$]

#math.equation(block: true, alt: "4 b minus 20")[$4 b − 20$]

#solutionbox[
4(#emph[b] − 5)
]

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

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

#solutionbox[
7(#emph[x] − 1)
]

#math.equation(block: true, alt: "5 m squared plus 20 m plus 35")[$5 m^(2) + 20 m + 35$]

#math.equation(block: true, alt: "3 n squared plus 21 n plus 12")[$3 n^(2) + 21 n + 12$]

#solutionbox[
3(#emph[n]#super[2] + 7#emph[n] + 4)
]

#math.equation(block: true, alt: "8 p squared plus 32 p plus 48")[$8 p^(2) + 32 p + 48$]

#math.equation(block: true, alt: "6 q squared plus 30 q plus 42")[$6 q^(2) + 30 q + 42$]

#solutionbox[
6(#emph[q]#super[2] + 5#emph[q] + 7)
]

#math.equation(block: true, alt: "8 q squared plus 15 q")[$8 q^(2) + 15 q$]

#math.equation(block: true, alt: "9 c squared plus 22 c")[$9 c^(2) + 22 c$]

#solutionbox[
#emph[c] (9#emph[c] + 22)
]

#math.equation(block: true, alt: "13 k squared plus 5 k")[$13 k^(2) + 5 k$]

#math.equation(block: true, alt: "17 x squared plus 7 x")[$17 x^(2) + 7 x$]

#solutionbox[
#emph[x] (17#emph[x] + 7)
]

#math.equation(block: true, alt: "5 c squared plus 9 c")[$5 c^(2) + 9 c$]

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

#solutionbox[
#emph[q] (4#emph[q] + 7)
]

#math.equation(block: true, alt: "5 p squared plus 25 p")[$5 p^(2) + 25 p$]

#math.equation(block: true, alt: "3 r squared plus 27 r")[$3 r^(2) + 27 r$]

#solutionbox[
3#emph[r] (#emph[r] + 9)
]

#math.equation(block: true, alt: "24 q squared minus 12 q")[$24 q^(2) − 12 q$]

#math.equation(block: true, alt: "30 u squared minus 10 u")[$30 u^(2) − 10 u$]

#solutionbox[
10#emph[u] (3#emph[u] − 1)
]

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

#math.equation(block: true, alt: "a b plus 8 b")[$a b + 8 b$]

#solutionbox[
#emph[b] (#emph[a] + 8)
]

#math.equation(block: true, alt: "60 x minus 6 x cubed")[$60 x − 6 x^(3)$]

#math.equation(block: true, alt: "55 y minus 11 y to the power 4")[$55 y − 11 y^(4)$]

#solutionbox[
11#emph[y] (5 − #emph[y]#super[3])
]

#math.equation(block: true, alt: "48 r to the power 4 minus 12 r cubed")[$48 r^(4) − 12 r^(3)$]

#math.equation(block: true, alt: "45 c cubed minus 15 c squared")[$45 c^(3) − 15 c^(2)$]

#solutionbox[
15#emph[c]#super[2] (3#emph[c] − 1)
]

#math.equation(block: true, alt: "4 a cubed minus 4 a b squared")[$4 a^(3) − 4 a b^(2)$]

#math.equation(block: true, alt: "6 c cubed minus 6 c d squared")[$6 c^(3) − 6 c d^(2)$]

#solutionbox[
6#emph[c] (#emph[c]#super[2] − #emph[d]#super[2])
]

#math.equation(block: true, alt: "30 u cubed plus 80 u squared")[$30 u^(3) + 80 u^(2)$]

#math.equation(block: true, alt: "48 x cubed plus 72 x squared")[$48 x^(3) + 72 x^(2)$]

#solutionbox[
24#emph[x]#super[2] (2#emph[x] + 3)
]

#math.equation(block: true, alt: "120 y to the power 6 plus 48 y to the power 4")[$120 y^(6) + 48 y^(4)$]

#math.equation(block: true, alt: "144 a to the power 6 plus 90 a cubed")[$144 a^(6) + 90 a^(3)$]

#solutionbox[
18#emph[a]#super[3] (8#emph[a]#super[3] + 5)
]

#math.equation(block: true, alt: "4 q squared plus 24 q plus 28")[$4 q^(2) + 24 q + 28$]

#math.equation(block: true, alt: "10 y squared plus 50 y plus 40")[$10 y^(2) + 50 y + 40$]

#solutionbox[
10(#emph[y]#super[2] + 5#emph[y] + 4)
]

#math.equation(block: true, alt: "15 z squared minus 30 z minus 90")[$15 z^(2) − 30 z − 90$]

#math.equation(block: true, alt: "12 u squared minus 36 u minus 108")[$12 u^(2) − 36 u − 108$]

#solutionbox[
12(#emph[u]#super[2] − 3#emph[u] − 9)
]

#math.equation(block: true, alt: "3 a to the power 4 minus 24 a cubed plus 18 a squared")[$3 a^(4) − 24 a^(3) + 18 a^(2)$]

#math.equation(block: true, alt: "5 p to the power 4 minus 20 p cubed minus 15 p squared")[$5 p^(4) − 20 p^(3) − 15 p^(2)$]

#solutionbox[
5#emph[p]#super[2] (#emph[p]#super[2] − 4#emph[p] − 3)
]

#math.equation(block: true, alt: "11 x to the power 6 plus 44 x to the power 5 minus 121 x to the power 4")[$11 x^(6) + 44 x^(5) − 121 x^(4)$]

#math.equation(block: true, alt: "8 c to the power 5 plus 40 c to the power 4 minus 56 c cubed")[$8 c^(5) + 40 c^(4) − 56 c^(3)$]

#solutionbox[
8#emph[c]#super[3] (#emph[c]#super[2] + 5#emph[c] − 7)
]

#math.equation(block: true, alt: "−3 n minus 24")[$−3 n − 24$]

#math.equation(block: true, alt: "−7 p minus 84")[$−7 p − 84$]

#solutionbox[
−7(#emph[p] + 12)
]

#math.equation(block: true, alt: "−15 a squared minus 40 a")[$−15 a^(2) − 40 a$]

#math.equation(block: true, alt: "−18 b squared minus 66 b")[$−18 b^(2) − 66 b$]

#solutionbox[
−6#emph[b] (3#emph[b] + 11)
]

#math.equation(block: true, alt: "−10 y cubed plus 60 y squared")[$−10 y^(3) + 60 y^(2)$]

#math.equation(block: true, alt: "−8 a cubed plus 32 a squared")[$−8 a^(3) + 32 a^(2)$]

#solutionbox[
−8#emph[a]#super[2] (#emph[a] − 4)
]

#math.equation(block: true, alt: "−4 u to the power 5 plus 56 u cubed")[$−4 u^(5) + 56 u^(3)$]

#math.equation(block: true, alt: "−9 b to the power 5 plus 63 b cubed")[$−9 b^(5) + 63 b^(3)$]

#solutionbox[
−9#emph[b]#super[3] (#emph[b]#super[2] − 7)
]


==== Everyday Math

#strong[Revenue] A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of #math.equation(block: false, alt: "p")[$p$] dollars each is given by the polynomial #math.equation(block: false, alt: "−5 p squared plus 150 p .")[$−5 p^(2) + 150 p .$] Factor the greatest common factor from this polynomial.

#strong[Height of a baseball] The height of a baseball hit with velocity #math.equation(block: false, alt: "80")[$80$] feet/second at #math.equation(block: false, alt: "4")[$4$] feet above ground level is #math.equation(block: false, alt: "−16 t squared plus 80 t plus 4 ,")[$−16 t^(2) + 80 t + 4 ,$] with #math.equation(block: false, alt: "t equals")[$t =$] the number of seconds since it was hit. Factor the greatest common factor from this polynomial.

#solutionbox[
−4(4#emph[t]#super[2] − 20#emph[t] − 1)
]


==== Writing Exercises

The greatest common factor of #math.equation(block: false, alt: "36")[$36$] and #math.equation(block: false, alt: "60")[$60$] is #math.equation(block: false, alt: "12 .")[$12 .$] Explain what this means.

What is the GCF of #math.equation(block: false, alt: "y to the power 4")[$y^(4)$], #math.equation(block: false, alt: "y to the power 5")[$y^(5)$], and #math.equation(block: false, alt: "y to the power 10")[$y^(10)$]? Write a general rule that tells how to find the GCF of #math.equation(block: false, alt: "y to the power a")[$y^("a")$], #math.equation(block: false, alt: "y to the power b")[$y^("b")$], and #math.equation(block: false, alt: "y to the power c")[$y^("c")$].

#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[A self-assessment table for math skills, with columns for 'Confidently,' 'With some help,' and 'No-I don't get it!' The skills listed are finding the greatest common factor and factoring it from a polynomial.], alt: "A self-assessment table for math skills, with columns for 'Confidently,' 'With some help,' and 'No-I don't get it!' The skills listed are finding the greatest common factor and factoring it from a polynomial.", caption: none)

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


=== Chapter Review Exercises


==== Add and Subtract Polynomials

#strong[Identify Polynomials, Monomials, Binomials and Trinomials]

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

#math.equation(block: true, alt: "y squared plus 8 y minus 20")[$y^(2) + 8 y − 20$]

#solutionbox[
trinomial
]

#math.equation(block: true, alt: "−6 a to the power 4")[$−6 a^(4)$]

#math.equation(block: true, alt: "9 x cubed minus 1")[$9 x^(3) − 1$]

#solutionbox[
binomial
]

#math.equation(block: true, alt: "n cubed minus 3 n squared plus 3 n minus 1")[$n^(3) − 3 n^(2) + 3 n − 1$]

#strong[Determine the Degree of Polynomials]

In the following exercises, determine the degree of each polynomial.

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

#solutionbox[
2
]

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

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

#solutionbox[
0
]

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

#strong[Add and Subtract Monomials]

In the following exercises, add or subtract the monomials.

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

#solutionbox[
15#emph[p]
]

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

Add #math.equation(block: false, alt: "4 n to the power 5 , − n to the power 5 , −6 n to the power 5")[$4 n^(5) , #h(0.2em) "−" n^(5) , #h(0.2em) −6 n^(5)$]

#solutionbox[
−3#emph[n]#super[5]
]

Subtract #math.equation(block: false, alt: "10 x squared")[$10 x^(2)$] from #math.equation(block: false, alt: "3 x squared")[$3 x^(2)$]

#strong[Add and Subtract Polynomials]

In the following exercises, add or subtract the polynomials.

#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")[$( 4 a^(2) + 9 a − 11 ) + ( 6 a^(2) − 5 a + 10 )$]

#solutionbox[
10#emph[a]#super[2] + 4#emph[a] − 1
]

#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")[$( 8 m^(2) + 12 m − 5 ) − ( 2 m^(2) − 7 m − 1 )$]

#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")[$( y^(2) − 3 y + 12 ) + ( 5 y^(2) − 9 )$]

#solutionbox[
6#emph[y]#super[2] − 3#emph[y] + 3
]

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

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")[$q^(2) + 6 q − 2$]

#solutionbox[
8#emph[q]#super[3] + #emph[q]#super[2] + 6#emph[q] − 29
]

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")[$x^(2) − 8 x + 15$]

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

In the following exercises, evaluate each polynomial for the given value.

#math.equation(block: false, alt: "200 x minus the fraction 1 over 5 x squared")[$200 x − frac(1, 5) #h(0.1em) x^(2)$] when #math.equation(block: false, alt: "x equals 5")[$x = 5$]

#solutionbox[
995
]

#math.equation(block: false, alt: "200 x minus the fraction 1 over 5 x squared")[$200 x − frac(1, 5) #h(0.1em) x^(2)$] when #math.equation(block: false, alt: "x equals 0")[$x = 0$]

#math.equation(block: false, alt: "200 x minus the fraction 1 over 5 x squared")[$200 x − frac(1, 5) #h(0.1em) x^(2)$] when #math.equation(block: false, alt: "x equals 15")[$x = 15$]

#solutionbox[
2,955
]

#math.equation(block: false, alt: "5 plus 40 x minus the fraction 1 over 2 x squared")[$5 + 40 x − frac(1, 2) #h(0.1em) x^(2)$] when #math.equation(block: false, alt: "x equals 10")[$x = 10$]

#math.equation(block: false, alt: "5 plus 40 x minus the fraction 1 over 2 x squared")[$5 + 40 x − frac(1, 2) #h(0.1em) x^(2)$] when #math.equation(block: false, alt: "x equals −4")[$x = −4$]

#solutionbox[
−163
]

#math.equation(block: false, alt: "5 plus 40 x minus the fraction 1 over 2 x squared")[$5 + 40 x − frac(1, 2) #h(0.1em) x^(2)$] when #math.equation(block: false, alt: "x equals 0")[$x = 0$]

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

#solutionbox[
64 feet
]

The fuel efficiency (in miles per gallon) of a bus going at a speed of #math.equation(block: false, alt: "x")[$x$] miles per hour is given by the polynomial #math.equation(block: false, alt: "minus the fraction 1 over 160 x squared plus the fraction 1 over 2 x .")[$− frac(1, 160) #h(0.1em) x^(2) + frac(1, 2) #h(0.1em) x .$] Find the fuel efficiency when #math.equation(block: false, alt: "x equals 20")[$x = 20$] mph.


==== Use Multiplication Properties of Exponents

#strong[Simplify Expressions with Exponents]

In the following exercises, simplify.

#math.equation(block: true, alt: "6 cubed")[$6^(3)$]

#solutionbox[
216
]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 2 close parenthesis to the power 4")[$attach(( frac(1, 2) ), t: 4)$]

#math.equation(block: true, alt: "open parenthesis −0.5 close parenthesis squared")[$attach(( −0.5 ), t: 2)$]

#solutionbox[
0.25
]

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

#strong[Simplify Expressions Using the Product Property of Exponents]

In the following exercises, simplify each expression.

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

#solutionbox[
#emph[p]#super[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[
#emph[a]#super[6]
]

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

#strong[Simplify Expressions Using the Power Property of Exponents]

In the following exercises, simplify each expression.

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

#solutionbox[
#emph[y]#super[12]
]

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

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

#solutionbox[
3#super[10]
]

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

#strong[Simplify Expressions Using the Product to a Power Property]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "open parenthesis 8 n close parenthesis squared")[$attach(( 8 n ), t: 2)$]

#solutionbox[
64#emph[n]#super[2]
]

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

#math.equation(block: true, alt: "open parenthesis 2 a b close parenthesis to the power 8")[$attach(( 2 a b ), t: 8)$]

#solutionbox[
256#emph[a]#super[8]#emph[b]#super[8]
]

#math.equation(block: true, alt: "open parenthesis −10 m n p close parenthesis to the power 4")[$attach(( −10 m n p ), t: 4)$]

#strong[Simplify Expressions by Applying Several Properties]

In the following exercises, simplify each expression.

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

#solutionbox[
27#emph[a]#super[15]
]

#math.equation(block: true, alt: "open parenthesis 4 y close parenthesis squared open parenthesis 8 y close parenthesis")[$attach(( 4 y ), t: 2) ( 8 y )$]

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

#solutionbox[
#emph[x]#super[21]
]

#math.equation(block: true, alt: "open parenthesis 5 s t squared close parenthesis cubed open parenthesis 2 s cubed t to the power 4 close parenthesis squared")[$attach(( 5 s t^(2) ), t: 3) attach(( 2 s^(3) t^(4) ), t: 2)$]

#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")[$( −6 p^(4) ) ( 9 p )$]

#solutionbox[
−54#emph[p]#super[5]
]

#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")[$( frac(1, 3) #h(0.1em) c^(2) ) ( 30 c^(8) )$]

#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")[$( 8 x^(2) y^(5) ) ( 7 x y^(6) )$]

#solutionbox[
56#emph[x]#super[3]#emph[y]#super[11]
]

#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")[$( frac(2, 3) #h(0.1em) m^(3) n^(6) ) ( frac(1, 6) #h(0.1em) m^(4) n^(4) )$]


==== Multiply Polynomials

#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")[$7 ( 10 − x )$]

#solutionbox[
70 − 7#emph[x]
]

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

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

#solutionbox[
−625#emph[y]#super[4] + 5#emph[y]
]

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

#strong[Multiply a Binomial by a Binomial]

In the following exercises, multiply the binomials using various methods.

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

#solutionbox[
#emph[a]#super[2] + 7#emph[a] + 10
]

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

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

#solutionbox[
6#emph[x]#super[2] − 19#emph[x] − 7
]

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

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

#solutionbox[
#emph[n]#super[2] + 9#emph[n] + 8
]

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

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

#solutionbox[
5#emph[u]#super[2] + 37#emph[u] − 24
]

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

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

#solutionbox[
#emph[p]#super[2] + 11#emph[p] + 28
]

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

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

#solutionbox[
27#emph[c]#super[2] − 3#emph[c] − 4
]

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

#strong[Multiply a Trinomial by a Binomial]

In the following exercises, multiply using any method.

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

#solutionbox[
#emph[x]#super[3] − 2#emph[x]#super[2] − 24#emph[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")[$( 5 b − 2 ) ( 3 b^(2) + b − 9 )$]

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

#solutionbox[
#emph[m]#super[3] − #emph[m]#super[2] − 72#emph[m] − 180
]

#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")[$( 4 y − 1 ) ( 6 y^(2) − 12 y + 5 )$]


==== Divide Monomials

#strong[Simplify Expressions Using the Quotient Property of Exponents]

In the following exercises, simplify.

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

#solutionbox[
2#super[6] #emph[or] 64
]

#math.equation(block: true, alt: "the fraction a to the power 6 over a")[$frac(#h(0.2em) a^(6), a)$]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over n to the power 9")[$frac(1, n^(9))$]
]

#math.equation(block: true, alt: "the fraction x over x to the power 5")[$frac(x, #h(0.2em) x^(5))$]

#strong[Simplify Expressions with Zero Exponents]

In the following exercises, simplify.

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

#solutionbox[
1
]

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

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

#solutionbox[
1
]

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

#strong[Simplify Expressions Using the Quotient to a Power Property]

In the following exercises, simplify.

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 over 25")[$frac(9, 25)$]
]

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

#math.equation(block: true, alt: "open parenthesis the fraction 5 m over n close parenthesis cubed")[$attach(( frac(5 m, n) ), t: 3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 125 m cubed over n cubed")[$frac(125 m^(3), n^(3))$]
]

#math.equation(block: true, alt: "open parenthesis the fraction s over 10 t close parenthesis squared")[$attach(( frac(s, 10 t) ), t: 2)$]

#strong[Simplify Expressions by Applying Several Properties]

In the following exercises, simplify.

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

#solutionbox[
#emph[a]#super[2]
]

#math.equation(block: true, alt: "the fraction u cubed over u squared times u to the power 4")[$frac(#h(0.2em) u^(3), u^(2) · u^(4))$]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over x to the power 40")[$frac(1, x^(40))$]
]

#math.equation(block: true, alt: "open parenthesis the fraction p to the power 4 times p to the power 5 over p cubed close parenthesis squared")[$attach(( frac(p^(4) · p^(5), p^(3)) ), t: 2)$]

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

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

#math.equation(block: true, alt: "open parenthesis the fraction 5 s squared over 4 t close parenthesis cubed")[$attach(( frac(5 s^(2), 4 t) ), t: 3)$]

#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[
9#emph[p]#super[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")[$frac(#h(0.2em) 45 y^(6), −15 y^(10))$]

#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")[$frac(#h(0.2em) 11 u^(6) v^(3), 55 u^(2) v^(8))$]

#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 42 r squared s to the power 4 over 6 r s cubed minus the fraction 54 r s squared over 9 s")[$frac(42 r^(2) s^(4), 6 r s^(3)) − frac(54 r s^(2), 9 s)$]


==== Integer Exponents and Scientific Notation

#strong[Use the Definition of a Negative Exponent]

In the following exercises, simplify.

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 36")[$frac(1, 36)$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 16")[$frac(5, 16)$]
]

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

#strong[Simplify Expressions with Integer Exponents]

In the following exercises, simplify.

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

#solutionbox[
#emph[x]#super[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: "open parenthesis k to the power −2 close parenthesis to the power −3")[$attach(( k^(−2) ), t: −3)$]

#solutionbox[
#emph[k]#super[6]
]

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

#math.equation(block: true, alt: "the fraction b to the power 8 over b to the power −2")[$frac(b^(8), #h(0.2em) b^(−2))$]

#solutionbox[
#emph[b]#super[10]
]

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

#strong[Convert from Decimal Notation to Scientific Notation]

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

#math.equation(block: true, alt: "5,300,000")[$5,300,000$]

#solutionbox[
5.3 × 10#super[6]
]

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

The thickness of a piece of paper is about #math.equation(block: false, alt: "0.097")[$0.097$] millimeter.

#solutionbox[
9.7 × 10#super[−2] millimeter
]

According to www.cleanair.com, U.S. businesses use about #math.equation(block: false, alt: "21,000,000")[$21,000,000$] tons of paper per year.

#strong[Convert Scientific Notation to Decimal Form]

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[
29,000
]

#math.equation(block: true, alt: "1.5 times 10 to the power 8")[$1.5 #h(0.2em) × #h(0.2em) 10^(8)$]

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

#solutionbox[
0.375
]

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

#strong[Multiply and Divide Using Scientific Notation]

In the following exercises, multiply and 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")[$( 3 #h(0.2em) × #h(0.2em) 10^(7) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−4) )$]

#solutionbox[
6,000
]

#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")[$( 1.5 #h(0.2em) × #h(0.2em) 10^(−3) ) ( 4.8 #h(0.2em) × #h(0.2em) 10^(−1) )$]

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

#solutionbox[
30,000,000,000
]

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


==== Introduction to Factoring Polynomials

#strong[Find the Greatest Common Factor of Two or More Expressions]

In the following exercises, find the greatest common factor.

#math.equation(block: true, alt: "5 n , 45")[$5 n , #h(0.2em) 45$]

#solutionbox[
5
]

#math.equation(block: true, alt: "8 a , 72")[$8 a , #h(0.2em) 72$]

#math.equation(block: true, alt: "12 x squared , 20 x cubed , 36 x to the power 4")[$12 x^(2) , #h(0.2em) 20 x^(3) , #h(0.2em) 36 x^(4)$]

#solutionbox[
4#emph[x]#super[2]
]

#math.equation(block: true, alt: "9 y to the power 4 , 21 y to the power 5 , 15 y to the power 6")[$9 y^(4) , #h(0.2em) 21 y^(5) , #h(0.2em) 15 y^(6)$]

#strong[Factor the Greatest Common Factor from a Polynomial]

In the following exercises, factor the greatest common factor from each polynomial.

#math.equation(block: true, alt: "16 u minus 24")[$16 u − 24$]

#solutionbox[
8(2#emph[u] − 3)
]

#math.equation(block: true, alt: "15 r plus 35")[$15 r + 35$]

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

#solutionbox[
6#emph[p] (#emph[p] + 1)
]

#math.equation(block: true, alt: "10 c squared minus 10 c")[$10 c^(2) − 10 c$]

#math.equation(block: true, alt: "−9 a to the power 5 minus 9 a cubed")[$−9 a^(5) − 9 a^(3)$]

#solutionbox[
−9#emph[a]#super[3] (#emph[a]#super[2] + 1)
]

#math.equation(block: true, alt: "−7 x to the power 8 minus 28 x cubed")[$−7 x^(8) − 28 x^(3)$]

#math.equation(block: true, alt: "5 y squared minus 55 y plus 45")[$5 y^(2) − 55 y + 45$]

#solutionbox[
5(#emph[y]#super[2] − 11#emph[y] + 9)
]

#math.equation(block: true, alt: "2 q to the power 5 minus 16 q cubed plus 30 q squared")[$2 q^(5) − 16 q^(3) + 30 q^(2)$]


=== 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
]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "open parenthesis 5 a squared plus 2 a minus 12 close parenthesis plus 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[
6#emph[x]#super[2] − 3#emph[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: "n times n to the power 4")[$n · n^(4)$]

#solutionbox[
#emph[n]#super[5]
]

#math.equation(block: true, alt: "open parenthesis 10 p cubed q to the power 5 close parenthesis squared")[$attach(( 10 p^(3) q^(5) ), t: 2)$]

#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[
−48#emph[x]#super[5]#emph[y]#super[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 s plus 8 close parenthesis open parenthesis s plus 9 close parenthesis")[$( s + 8 ) ( s + 9 )$]

#solutionbox[
#emph[s]#super[2] + 17#emph[s] + 72
]

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

#math.equation(block: true, alt: "open parenthesis 11 a minus 6 close parenthesis open parenthesis 5 a minus 1 close parenthesis")[$( 11 a − 6 ) ( 5 a − 1 )$]

#solutionbox[
55#emph[a]#super[2] − 41#emph[a] + 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 a plus 9 b close parenthesis open parenthesis 6 a minus 5 b close parenthesis")[$( 4 a + 9 b ) ( 6 a − 5 b )$]

#solutionbox[
24#emph[a]#super[2] + 34#emph[ab] − 45#emph[b]#super[2]
]

#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 the fraction x cubed times x to the power 9 over x to the power 5 close parenthesis squared")[$attach(( frac(x^(3) · x^(9), x^(5)) ), t: 2)$]

#solutionbox[
#emph[x]#super[14]
]

#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")[$attach(( 47 a^(18) b^(23) c^(5) ), t: 0)$]

#math.equation(block: true, alt: "the fraction 24 r cubed s over 6 r squared s to the power 7")[$frac(24 r^(3) s, #h(0.2em) 6 r^(2) s^(7))$]

#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: "the fraction 8 y squared minus 16 y plus 20 over 4 y")[$frac(8 y^(2) − 16 y + 20, 4 y)$]

#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[
3#emph[y]#super[2] − 7#emph[x]
]

#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")[$frac(x^(4), #h(0.2em) x^(−5))$]

#solutionbox[
#emph[x]#super[9]
]

In the following exercises, factor the greatest common factor from each polynomial.

#math.equation(block: true, alt: "80 a cubed plus 120 a squared plus 40 a")[$80 a^(3) + 120 a^(2) + 40 a$]

#math.equation(block: true, alt: "−6 x squared minus 30 x")[$−6 x^(2) − 30 x$]

#solutionbox[
−6#emph[x] (#emph[x] + 5)
]

According to www.cleanair.org, the amount of trash generated in the US in one year averages out to #math.equation(block: false, alt: "112,000")[$112,000$] pounds of trash per person. Write this number in 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.

#solutionbox[
0.000525
]

In the following exercises, simplify, and write your answer in decimal form.

#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")[$( 2.4 #h(0.2em) × #h(0.2em) 10^(8) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−5) )$]

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

#solutionbox[
#math.equation(block: true, alt: "300,000")[$300,000$]
]

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