#set document(title: "8.2 Multiply and Divide Rational Expressions", author: "OpenStax / XYZ Homework")
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== 8.2#h(0.6em)Multiply and Divide Rational Expressions

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

If you miss a problem, go back to the section listed and review the material.

Multiply: #math.equation(block: false, alt: "the fraction 14 over 15 times the fraction 6 over 35 .")[$frac(14, 15) · frac(6, 35) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 25")[$frac(4, 25)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 14 over 15 divided by the fraction 6 over 35 .")[$frac(14, 15) ÷ frac(6, 35) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 49 over 9")[$frac(49, 9)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "2 x squared minus 98 .")[$2 x^(2) − 98 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 x minus 7 x plus 7")[$2 x − 7 x + 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "10 n cubed plus 10 .")[$10 n^(3) + 10 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "10 n plus 1 n squared minus n plus 1")[$10 n + 1 n^(2) − n + 1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "10 p squared minus 25 p q minus 15 q squared .")[$10 p^(2) − 25 p q − 15 q^(2) .$] #linebreak() 
If you missed this problem, review .

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


=== Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplication of Rational Expressions]

If #math.equation(block: false, alt: "p , q , r , s")[$p , q , r , s$] are polynomials where #math.equation(block: false, alt: "q not equal to 0 and s not equal to 0")[$q ≠ 0 #h(0.2em) "and" #h(0.2em) s ≠ 0$], then

#math.equation(block: true, alt: "the fraction p over q times the fraction r over s equals the fraction p r over q s")[$frac(p, q) · frac(r, s) = frac(p r, q s)$]To multiply rational expressions, multiply the numerators and multiply the denominators.
]

We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.

#examplebox("Example 1")[][
Multiply: #math.equation(block: false, alt: "the fraction 10 over 28 times the fraction 8 over 15 .")[$frac(10, 28) · frac(8, 15) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the multiplication of two fractions: 10/28 multiplied by 8/15, represented as (10/28)   (8/15).], alt: "The image displays the multiplication of two fractions: 10/28 multiplied by 8/15, represented as (10/28) (8/15).", caption: none)]),
  [Multiply the numerators and denominators.], [#figure(figph[A mathematical expression displaying the fraction (10 multiplied by 8) divided by (28 multiplied by 15).], alt: "A mathematical expression displaying the fraction (10 multiplied by 8) divided by (28 multiplied by 15).", caption: none)],
  [Look for common factors, and then remove them.], [#figure(figph[A mathematical fraction is shown with terms being cancelled out. In the numerator, 5 and 4 are crossed out, leaving 2 multiplied by 2. In the denominator, 4 and 5 are crossed out, leaving 7 multiplied by 3.], alt: "A mathematical fraction is shown with terms being cancelled out. In the numerator, 5 and 4 are crossed out, leaving 2 multiplied by 2. In the denominator, 4 and 5 are crossed out, leaving 7 multiplied by 3.", caption: none)],
  [Simplify.], [#figure(figph[The fraction 4/21 is displayed on a white background, with 4 as the numerator and 21 as the denominator, separated by a horizontal line.], alt: "The fraction 4/21 is displayed on a white background, with 4 as the numerator and 21 as the denominator, separated by a horizontal line.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Mulitply: #math.equation(block: false, alt: "the fraction 6 over 10 times the fraction 15 over 12 .")[$frac(6, 10) · frac(15, 12) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Mulitply: #math.equation(block: false, alt: "the fraction 20 over 15 times the fraction 6 over 8 .")[$frac(20, 15) · frac(6, 8) .$]

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

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, #math.equation(block: false, alt: "x not equal to 0")[$x ≠ 0$] and #math.equation(block: false, alt: "y not equal to 0")[$y ≠ 0$].

#examplebox("Example 2")[][
Mulitply: #math.equation(block: false, alt: "the fraction 2 x over 3 y squared times the fraction 6 x y cubed over x squared y .")[$frac(2 x, 3 y^(2)) · frac(6 x y^(3), x^(2) y) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the product of two fractions: (2x / 3y^2) multiplied by (6xy^3 / x^2y).], alt: "A mathematical expression showing the product of two fractions: (2x / 3y^2) multiplied by (6xy^3 / x^2y).", caption: none)]),
  [Multiply.], [#figure(figph[An algebraic fraction is displayed with the numerator '2x multiplied by 6xy^3' and the denominator '3y^2 multiplied by x^2y'.], alt: "An algebraic fraction is displayed with the numerator '2x multiplied by 6xy^3' and the denominator '3y^2 multiplied by x^2y'.", caption: none)],
  [Factor the numerator and denominator completely, and then remove common factors.], [#figure(figph[A mathematical fraction showing simplification. Common factors like 'x', 'B', and 'y' are struck through in red from both the numerator and denominator, illustrating term cancellation.], alt: "A mathematical fraction showing simplification. Common factors like 'x', 'B', and 'y' are struck through in red from both the numerator and denominator, illustrating term cancellation.", caption: none)],
  [Simplify.], [#figure(figph[The number 4 in black against a white background.], alt: "The number 4 in black against a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Mulitply: #math.equation(block: false, alt: "the fraction 3 p q over q squared times the fraction 5 p squared q over 6 p q .")[$frac(3 p q, q^(2)) · frac(5 p^(2) q, 6 p q) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 p squared over 2 q")[$frac(5 p^(2), 2 q)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Mulitply: #math.equation(block: false, alt: "the fraction 6 x cubed y over 7 x squared times the fraction 2 x y cubed over x squared y .")[$frac(6 x^(3) y, 7 x^(2)) · frac(2 x y^(3), x^(2) y) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 12 y cubed over 7")[$frac(12 y^(3), 7)$]
]
]

#examplebox("Example 3")[How to Multiply Rational Expressions][
Mulitply: #math.equation(block: false, alt: "the fraction 2 x over x squared minus 7 x plus 12 times the fraction x squared minus 9 over 6 x squared .")[$frac(2 x, x^(2) − 7 x + 12) · frac(x^(2) − 9, 6 x^(2)) .$]

#solutionbox[
#figure(figph[The above image has three columns and three rows to show how to multiply rational expressions. Step one is to factor each numerator and denominator completely. Factor x squared minus 9 and x squared minus 7 x plus 12. The rational equation is 2x divided by x squared plus x plus 12 times x squared minus 9 divided by 6x squared, then to 2x divided by x minus 3 times x minus 4 times x minus 3 times x plus 3 divided by 6x squared.], alt: "The above image has three columns and three rows to show how to multiply rational expressions. Step one is to factor each numerator and denominator completely. Factor x squared minus 9 and x squared minus 7 x plus 12. The rational equation is 2x divided by x squared plus x plus 12 times x squared minus 9 divided by 6x squared, then to 2x divided by x minus 3 times x minus 4 times x minus 3 times x plus 3 divided by 6x squared.", caption: none)

#figure(figph[Step 2 is to multiply the numerators and denominators. It is helpful to multiply the monomials first. Multiply 2x times x minus 3 times x plus 3 divided by 6x squared times x minus 3 times x minus 4.], alt: "Step 2 is to multiply the numerators and denominators. It is helpful to multiply the monomials first. Multiply 2x times x minus 3 times x plus 3 divided by 6x squared times x minus 3 times x minus 4.", caption: none)

#figure(figph[Step 3 is to divide out the common factors, canceling out 2, x, and x minus 3 in the numerator and 2, x and x minus 3 in the denominator. Leave the denominator in factored form to get x plus 3 divided by 3x times x minus 4.], alt: "Step 3 is to divide out the common factors, canceling out 2, x, and x minus 3 in the numerator and 2, x and x minus 3 in the denominator. Leave the denominator in factored form to get x plus 3 divided by 3x times x minus 4.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Mulitply: #math.equation(block: false, alt: "the fraction 5 x over x squared plus 5 x plus 6 times the fraction x squared minus 4 over 10 x .")[$frac(5 x, x^(2) + 5 x + 6) · frac(x^(2) − 4, 10 x) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Mulitply: #math.equation(block: false, alt: "the fraction 9 x squared over x squared plus 11 x plus 30 times the fraction x squared minus 36 over 3 x squared .")[$frac(9 x^(2), x^(2) + 11 x + 30) · frac(x^(2) − 36, 3 x^(2)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 open parenthesis x minus 6 close parenthesis over x plus 5")[$frac(3 ( x − 6 ), x + 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiply a rational expression.]

+ Factor each numerator and denominator completely.
+ Multiply the numerators and denominators.
+ Simplify by dividing out common factors.
]

#examplebox("Example 4")[][
Multiply: #math.equation(block: false, alt: "the fraction n squared minus 7 n over n squared plus 2 n plus 1 times the fraction n plus 1 over 2 n .")[$frac(n^(2) − 7 n, n^(2) + 2 n + 1) · frac(n + 1, 2 n) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction n squared minus 7 n over n squared plus 2 n plus 1 times the fraction n plus 1 over 2 n")[$frac(n^(2) − 7 n, n^(2) + 2 n + 1) · frac(n + 1, 2 n)$]]),
  [Factor each numerator and denominator.], [#math.equation(block: false, alt: "the fraction n open parenthesis n minus 7 close parenthesis over open parenthesis n plus 1 close parenthesis open parenthesis n plus 1 close parenthesis times the fraction n plus 1 over 2 n")[$frac(n ( n − 7 ), ( n + 1 ) ( n + 1 )) · frac(n + 1, 2 n)$]],
  [Multiply the numerators and the denominators.], [#math.equation(block: false, alt: "the fraction n open parenthesis n minus 7 close parenthesis open parenthesis n plus 1 close parenthesis over open parenthesis n plus 1 close parenthesis open parenthesis n plus 1 close parenthesis 2 n")[$frac(n ( n − 7 ) ( n + 1 ), ( n + 1 ) ( n + 1 ) 2 n)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction n open parenthesis n minus 7 close parenthesis open parenthesis n plus 1 close parenthesis over open parenthesis n plus 1 close parenthesis open parenthesis n plus 1 close parenthesis 2 n")[$frac(cancel(n) ( n − 7 ) cancel(( n + 1 )), ( n + 1 ) cancel(( n + 1 )) 2 cancel(n))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction n minus 7 over 2 open parenthesis n plus 1 close parenthesis")[$frac(n − 7, 2 ( n + 1 ))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction x squared minus 25 over x squared minus 3 x minus 10 times the fraction x plus 2 over x .")[$frac(x^(2) − 25, x^(2) − 3 x − 10) · frac(x + 2, x) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction x squared minus 4 x over x squared plus 5 x plus 6 times the fraction x plus 2 over x .")[$frac(x^(2) − 4 x, x^(2) + 5 x + 6) · frac(x + 2, x) .$]

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

#examplebox("Example 5")[][
Multiply: #math.equation(block: false, alt: "the fraction 16 minus 4 x over 2 x minus 12 times the fraction x squared minus 5 x minus 6 over x squared minus 16 .")[$frac(16 − 4 x, 2 x − 12) · frac(x^(2) − 5 x − 6, x^(2) − 16) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 16 minus 4 x over 2 x minus 12 times the fraction x squared minus 5 x minus 6 over x squared minus 16")[$frac(16 − 4 x, 2 x − 12) · frac(x^(2) − 5 x − 6, x^(2) − 16)$]]),
  [Factor each numerator and denominator.], [#math.equation(block: false, alt: "the fraction 4 open parenthesis 4 minus x close parenthesis over 2 open parenthesis x minus 6 close parenthesis times the fraction open parenthesis x minus 6 close parenthesis open parenthesis x plus 1 close parenthesis over open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis")[$frac(4 ( 4 − x ), 2 ( x − 6 )) · frac(( x − 6 ) ( x + 1 ), ( x − 4 ) ( x + 4 ))$]],
  [Multiply the numerators and the denominators.], [#math.equation(block: false, alt: "the fraction 4 open parenthesis 4 minus x close parenthesis open parenthesis x minus 6 close parenthesis open parenthesis x plus 1 close parenthesis over 2 open parenthesis x minus 6 close parenthesis open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis")[$frac(4 ( 4 − x ) ( x − 6 ) ( x + 1 ), 2 ( x − 6 ) ( x − 4 ) ( x + 4 ))$]],
  [Remove common factors.], [#math.equation(block: false, alt: "open parenthesis −1 close parenthesis the fraction 2 times 2 open parenthesis 4 minus x close parenthesis open parenthesis x minus 6 close parenthesis open parenthesis x plus 1 close parenthesis over 2 open parenthesis x minus 6 close parenthesis open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis")[$( −1 ) frac(cancel(2) · 2 cancel(( 4 − x )) cancel(( x − 6 )) ( x + 1 ), cancel(2) cancel(( x − 6 )) cancel(( x − 4 )) ( x + 4 ))$]],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 2 open parenthesis x plus 1 close parenthesis over open parenthesis x plus 4 close parenthesis")[$− frac(2 ( x + 1 ), ( x + 4 ))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction 12 x minus 6 x squared over x squared plus 8 x times the fraction x squared plus 11 x plus 24 over x squared minus 4 .")[$frac(12 x − 6 x^(2), x^(2) + 8 x) · frac(x^(2) + 11 x + 24, x^(2) − 4) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 6 open parenthesis x plus 3 close parenthesis over x plus 2")[$− frac(6 ( x + 3 ), x + 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction 9 v minus 3 v squared over 9 v plus 36 times the fraction v squared plus 7 v plus 12 over v squared minus 9 .")[$frac(9 v − 3 v^(2), 9 v + 36) · frac(v^(2) + 7 v + 12, v^(2) − 9) .$]

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

#examplebox("Example 6")[][
Multiply: #math.equation(block: false, alt: "the fraction 2 x minus 6 over x squared minus 8 x plus 15 times the fraction x squared minus 25 over 2 x plus 10 .")[$frac(2 x − 6, x^(2) − 8 x + 15) · frac(x^(2) − 25, 2 x + 10) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the multiplication of two algebraic fractions: (2x-6)/(x^2-8x+15) \* (x^2-25)/(2x+10).], alt: "A mathematical expression showing the multiplication of two algebraic fractions: (2x-6)/(x^2-8x+15) * (x^2-25)/(2x+10).", caption: none)]),
  [Factor each numerator and denominator.], [#figure(figph[An algebraic expression illustrating the multiplication of two rational fractions: 2(x-3) over (x-3)(x-5) multiplied by (x-5)(x+5) over 2(x+5).], alt: "An algebraic expression illustrating the multiplication of two rational fractions: 2(x-3) over (x-3)(x-5) multiplied by (x-5)(x+5) over 2(x+5).", caption: none)],
  [Multiply the numerators and denominators.], [#figure(figph[The image displays a rational expression where the numerator and denominator are identical: 2(x-3)(x-5)(x+5) divided by 2(x-3)(x-5)(x+5). This expression simplifies to 1.], alt: "The image displays a rational expression where the numerator and denominator are identical: 2(x-3)(x-5)(x+5) divided by 2(x-3)(x-5)(x+5). This expression simplifies to 1.", caption: none)],
  [Remove common factors.], [#figure(figph[A fraction displays identical algebraic expressions in the numerator and denominator, with Z, (x-3), (x-5), and (x+5) terms all struck through, demonstrating cancellation.], alt: "A fraction displays identical algebraic expressions in the numerator and denominator, with Z, (x-3), (x-5), and (x+5) terms all struck through, demonstrating cancellation.", caption: none)],
  [Simplify.], [#figure(figph[1], alt: "1", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction 3 a minus 21 over a squared minus 9 a plus 14 times the fraction a squared minus 4 over 3 a plus 6 .")[$frac(3 a − 21, a^(2) − 9 a + 14) · frac(a^(2) − 4, 3 a + 6) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction b squared minus b over b squared plus 9 b minus 10 times the fraction b squared minus 100 over b squared minus 10 b .")[$frac(b^(2) − b, b^(2) + 9 b − 10) · frac(b^(2) − 100, b^(2) − 10 b) .$]

#solutionbox[
1
]
]


=== Divide Rational Expressions

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.

Remember, the #strong[reciprocal] of #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] is #math.equation(block: false, alt: "the fraction b over a")[$frac(b, a)$]. To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.

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

If #math.equation(block: false, alt: "p , q , r , s")[$p , q , r , s$] are polynomials where #math.equation(block: false, alt: "q not equal to 0 , r not equal to 0 , s not equal to 0")[$q ≠ 0 , r ≠ 0 , s ≠ 0$], then

#math.equation(block: true, alt: "the fraction p over q divided by the fraction r over s equals the fraction p over q times the fraction s over r")[$frac(p, q) ÷ frac(r, s) = frac(p, q) · frac(s, r)$]To divide rational expressions multiply the first fraction by the reciprocal of the second.
]

#examplebox("Example 7")[How to Divide Rational Expressions][
Divide: #math.equation(block: false, alt: "the fraction x plus 9 over 6 minus x divided by the fraction x squared minus 81 over x minus 6 .")[$frac(x + 9, 6 − x) ÷ frac(x^(2) − 81, x − 6) .$]

#solutionbox[
#figure(figph[The above image has three columns. It shows the steps to divide rational expressions. Step one is to rewrite the division as the product of the first rational expression and the reciprocal of the second for x plus 9 divided by 6 minus x divided by x squared minus 81 divided by x minus 6. “Flip” the second fraction and change the division sign to multiplication to get x plus 9 divided by 6 minus x times x minus 6 divided by x squared minus 81.], alt: "The above image has three columns. It shows the steps to divide rational expressions. Step one is to rewrite the division as the product of the first rational expression and the reciprocal of the second for x plus 9 divided by 6 minus x divided by x squared minus 81 divided by x minus 6. “Flip” the second fraction and change the division sign to multiplication to get x plus 9 divided by 6 minus x times x minus 6 divided by x squared minus 81.", caption: none)

#figure(figph[Step two is to factor the numerators and denominators completely. Factor x squared minus 81 to get x plus 9 divided by 6 minus x times x minus 6 divided by x minus 9 times x plus 9.], alt: "Step two is to factor the numerators and denominators completely. Factor x squared minus 81 to get x plus 9 divided by 6 minus x times x minus 6 divided by x minus 9 times x plus 9.", caption: none)

#figure(figph[Step three is to multiply the numerators and denominators to get x plus 9 times x minus 6 divided by 6 minus x times x minus 9 times x plus 9.], alt: "Step three is to multiply the numerators and denominators to get x plus 9 times x minus 6 divided by 6 minus x times x minus 9 times x plus 9.", caption: none)

#figure(figph[Step four is to simplify by dividing out common factors. Divide out the common factors x plus 9, x minus 6 from the numerator and 6 minus x and x plus 9 from the denominator. Remember opposites divide to negative 1. This simplifies to negative 1 divided by x minus 9.], alt: "Step four is to simplify by dividing out common factors. Divide out the common factors x plus 9, x minus 6 from the numerator and 6 minus x and x plus 9 from the denominator. Remember opposites divide to negative 1. This simplifies to negative 1 divided by x minus 9.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction c plus 3 over 5 minus c divided by the fraction c squared minus 9 over c minus 5 .")[$frac(c + 3, 5 − c) ÷ frac(c^(2) − 9, c − 5) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 2 minus d over d minus 4 divided by the fraction 4 minus d squared over 4 minus d .")[$frac(2 − d, d − 4) ÷ frac(4 − d^(2), 4 − d) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Divide rational expressions.]

+ Rewrite the division as the product of the first rational expression and the reciprocal of the second.
+ Factor the numerators and denominators completely.
+ Multiply the numerators and denominators together.
+ Simplify by dividing out common factors.
]

#examplebox("Example 8")[][
Divide: #math.equation(block: false, alt: "the fraction 3 n squared over n squared minus 4 n divided by the fraction 9 n squared minus 45 n over n squared minus 7 n plus 10 .")[$frac(3 n^(2), n^(2) − 4 n) ÷ frac(9 n^(2) − 45 n, n^(2) − 7 n + 10) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the division of two algebraic fractions. The first fraction is 3n^2 over n^2 - 4n, and the second is 9n^2 - 45n over n^2 - 7n + 10.], alt: "A mathematical expression showing the division of two algebraic fractions. The first fraction is 3n^2 over n^2 - 4n, and the second is 9n^2 - 45n over n^2 - 7n + 10.", caption: none)]),
  [Rewrite the division as the product of the first rational
expression and the reciprocal of the second.], [#figure(figph[A mathematical problem showing the multiplication of two rational expressions. The first expression is 3n^2 divided by n^2 minus 4n. The second expression is n^2 minus 7n plus 10 divided by 9n^2 minus 45n.], alt: "A mathematical problem showing the multiplication of two rational expressions. The first expression is 3n^2 divided by n^2 minus 4n. The second expression is n^2 minus 7n plus 10 divided by 9n^2 minus 45n.", caption: none)],
  [Factor the numerators and denominators
and then multiply.], [#figure(figph[A mathematical expression showing a fraction. The numerator is 3 \* n \* n \* (n - 5)(n - 2). The denominator is n(n - 4) \* 3 \* 3 \* n \* (n - 5).], alt: "A mathematical expression showing a fraction. The numerator is 3 * n * n * (n - 5)(n - 2). The denominator is n(n - 4) * 3 * 3 * n * (n - 5).", caption: none)],
  [Simplify by dividing out common factors.], [#figure(figph[An algebraic fraction undergoing simplification, with identical terms 'B', 'n', and 'n(n-5)' struck through in red in both the numerator and denominator.], alt: "An algebraic fraction undergoing simplification, with identical terms 'B', 'n', and 'n(n-5)' struck through in red in both the numerator and denominator.", caption: none)],
  [], [#figure(figph[A mathematical expression showing the fraction (n-2) divided by 3(n-4).], alt: "A mathematical expression showing the fraction (n-2) divided by 3(n-4).", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 2 m squared over m squared minus 8 m divided by the fraction 8 m squared plus 24 m over m squared plus m minus 6 .")[$frac(2 m^(2), m^(2) − 8 m) ÷ frac(8 m^(2) + 24 m, m^(2) + m − 6) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis m minus 2 close parenthesis over 4 open parenthesis m minus 8 close parenthesis")[$frac(( m − 2 ), 4 ( m − 8 ))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 15 n squared over 3 n squared plus 33 n divided by the fraction 5 n minus 5 over n squared plus 9 n minus 22 .")[$frac(15 n^(2), 3 n^(2) + 33 n) ÷ frac(5 n − 5, n^(2) + 9 n − 22) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n open parenthesis n minus 2 close parenthesis over n minus 1")[$frac(n ( n − 2 ), n − 1)$]
]
]

Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.

#examplebox("Example 9")[][
Divide: #math.equation(block: false, alt: "the fraction 2 x squared plus 5 x minus 12 over x squared minus 16 divided by the fraction 2 x squared minus 13 x plus 15 over x squared minus 8 x plus 16 .")[$frac(2 x^(2) + 5 x − 12, x^(2) − 16) ÷ frac(2 x^(2) − 13 x + 15, x^(2) − 8 x + 16) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 2 x squared plus 5 x minus 12 over x squared minus 16 divided by the fraction 2 x squared minus 13 x plus 15 over x squared minus 8 x plus 16")[$frac(2 x^(2) + 5 x − 12, x^(2) − 16) ÷ frac(2 x^(2) − 13 x + 15, x^(2) − 8 x + 16)$]]),
  [Rewrite the division as multiplication of #linebreak() the first expression by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 2 x squared plus 5 x minus 12 over x squared minus 16 times the fraction x squared minus 8 x plus 16 over 2 x squared minus 13 x plus 15")[$frac(2 x^(2) + 5 x − 12, x^(2) − 16) · frac(x^(2) − 8 x + 16, 2 x^(2) − 13 x + 15)$]],
  [Factor the numerators and denominators and then multiply.], [#math.equation(block: false, alt: "the fraction open parenthesis 2 x minus 3 close parenthesis open parenthesis x plus 4 close parenthesis open parenthesis x minus 4 close parenthesis open parenthesis x minus 4 close parenthesis over open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis open parenthesis 2 x minus 3 close parenthesis open parenthesis x minus 5 close parenthesis")[$frac(( 2 x − 3 ) ( x + 4 ) ( x − 4 ) ( x − 4 ), ( x − 4 ) ( x + 4 ) ( 2 x − 3 ) ( x − 5 ))$]],
  [Simplify by dividing out common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis 2 x minus 3 close parenthesis open parenthesis x plus 4 close parenthesis open parenthesis x minus 4 close parenthesis open parenthesis x minus 4 close parenthesis over open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis open parenthesis 2 x minus 3 close parenthesis open parenthesis x minus 5 close parenthesis")[$frac(cancel(( 2 x − 3 )) cancel(( x + 4 )) cancel(( x − 4 )) ( x − 4 ), cancel(( x − 4 )) cancel(( x + 4 )) cancel(( 2 x − 3 )) ( x − 5 ))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction x minus 4 over x minus 5")[$frac(x − 4, x − 5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 3 a squared minus 8 a minus 3 over a squared minus 25 divided by the fraction 3 a squared minus 14 a minus 5 over a squared plus 10 a plus 25 .")[$frac(3 a^(2) − 8 a − 3, a^(2) − 25) ÷ frac(3 a^(2) − 14 a − 5, a^(2) + 10 a + 25) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 4 b squared plus 7 b minus 2 over 1 minus b squared divided by the fraction 4 b squared plus 15 b minus 4 over b squared minus 2 b plus 1 .")[$frac(4 b^(2) + 7 b − 2, 1 − b^(2)) ÷ frac(4 b^(2) + 15 b − 4, b^(2) − 2 b + 1) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction open parenthesis b plus 2 close parenthesis open parenthesis b minus 1 close parenthesis over open parenthesis 1 plus b close parenthesis open parenthesis b plus 4 close parenthesis")[$− frac(( b + 2 ) ( b − 1 ), ( 1 + b ) ( b + 4 ))$]
]
]

#examplebox("Example 10")[][
Divide: #math.equation(block: false, alt: "the fraction p cubed plus q cubed over 2 p squared plus 2 p q plus 2 q squared divided by the fraction p squared minus q squared over 6 .")[$frac(p^(3) + q^(3), 2 p^(2) + 2 p q + 2 q^(2)) ÷ frac(p^(2) − q^(2), 6) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction p cubed plus q cubed over 2 p squared plus 2 p q plus 2 q squared divided by the fraction p squared minus q squared over 6")[$frac(p^(3) + q^(3), 2 p^(2) + 2 p q + 2 q^(2)) ÷ frac(p^(2) − q^(2), 6)$]]),
  [Rewrite the division as a multiplication #linebreak() of the first expression times the #linebreak() reciprocal of the second.], [#math.equation(block: false, alt: "the fraction p cubed plus q cubed over 2 p squared plus 2 p q plus 2 q squared times the fraction 6 over p squared minus q squared")[$frac(p^(3) + q^(3), 2 p^(2) + 2 p q + 2 q^(2)) · frac(6, p^(2) − q^(2))$]],
  [Factor the numerators and denominators and then multiply.], [#math.equation(block: false, alt: "the fraction open parenthesis p plus q close parenthesis open parenthesis p squared minus p q plus q squared close parenthesis 6 over 2 open parenthesis p squared plus p q plus q squared close parenthesis open parenthesis p minus q close parenthesis open parenthesis p plus q close parenthesis")[$frac(( p + q ) ( p^(2) − p q + q^(2) ) 6, 2 ( p^(2) + p q + q^(2) ) ( p − q ) ( p + q ))$]],
  [Simplify by dividing out common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis p plus q close parenthesis open parenthesis p squared minus p q plus q squared close parenthesis 6 cubed over 2 open parenthesis p squared plus p q plus q squared close parenthesis open parenthesis p minus q close parenthesis open parenthesis p plus q close parenthesis")[$frac(cancel(( p + q )) ( p^(2) − p q + q^(2) ) cancel(6)^(3), cancel(2) ( p^(2) + p q + q^(2) ) ( p − q ) cancel(( p + q )))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 open parenthesis p squared minus p q plus q squared close parenthesis over open parenthesis p minus q close parenthesis open parenthesis p squared plus p q plus q squared close parenthesis")[$frac(3 ( p^(2) − p q + q^(2) ), ( p − q ) ( p^(2) + p q + q^(2) ))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction x cubed minus 8 over 3 x squared minus 6 x plus 12 divided by the fraction x squared minus 4 over 6 .")[$frac(x^(3) − 8, 3 x^(2) − 6 x + 12) ÷ frac(x^(2) − 4, 6) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 2 z squared over z squared minus 1 divided by the fraction z cubed minus z squared plus z over z cubed minus 1 .")[$frac(2 z^(2), z^(2) − 1) ÷ frac(z^(3) − z^(2) + z, z^(3) − 1) .$]

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

Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide #math.equation(block: false, alt: "the fraction 3 over 5 divided by 4")[$frac(3, 5) ÷ 4$], we first write 4 as a fraction so that we can find its reciprocal.

#math.equation(block: true, alt: "the fraction 3 over 5 divided by 4; the fraction 3 over 5 divided by the fraction 4 over 1; the fraction 3 over 5 times the fraction 1 over 4")[$frac(3, 5) ÷ 4 \ frac(3, 5) ÷ frac(4, 1) \ frac(3, 5) · frac(1, 4)$]We do the same thing when we divide rational expressions.

#examplebox("Example 11")[][
Divide: #math.equation(block: false, alt: "the fraction a squared minus b squared over 3 a b divided by open parenthesis a squared plus 2 a b plus b squared close parenthesis .")[$frac(a^(2) − b^(2), 3 a b) ÷ ( a^(2) + 2 a b + b^(2) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction a squared minus b squared over 3 a b divided by open parenthesis a squared plus 2 a b plus b squared close parenthesis")[$frac(a^(2) − b^(2), 3 a b) ÷ ( a^(2) + 2 a b + b^(2) )$]]),
  [Write the second expression as a fraction.], [#math.equation(block: false, alt: "the fraction a squared minus b squared over 3 a b divided by the fraction a squared plus 2 a b plus b squared over 1")[$frac(a^(2) − b^(2), 3 a b) ÷ frac(a^(2) + 2 a b + b^(2), 1)$]],
  [Rewrite the division as the first #linebreak() expression times the reciprocal of the #linebreak() second expression.], [#math.equation(block: false, alt: "the fraction a squared minus b squared over 3 a b times the fraction 1 over a squared plus 2 a b plus b squared")[$frac(a^(2) − b^(2), 3 a b) · frac(1, a^(2) + 2 a b + b^(2))$]],
  [Factor the numerators and the #linebreak() denominators, and then multiply.], [#math.equation(block: false, alt: "the fraction open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis times 1 over 3 a b times open parenthesis a plus b close parenthesis open parenthesis a plus b close parenthesis")[$frac(( a − b ) ( a + b ) · 1, 3 a b · ( a + b ) ( a + b ))$]],
  [Simplify by dividing out common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis over 3 a b times open parenthesis a plus b close parenthesis open parenthesis a plus b close parenthesis")[$frac(( a − b ) cancel(( a + b )), 3 a b · cancel(( a + b )) ( a + b ))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction open parenthesis a minus b close parenthesis over 3 a b open parenthesis a plus b close parenthesis")[$frac(( a − b ), 3 a b ( a + b ))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 2 x squared minus 14 x minus 16 over 4 divided by open parenthesis x squared plus 2 x plus 1 close parenthesis .")[$frac(2 x^(2) − 14 x − 16, 4) ÷ ( x^(2) + 2 x + 1 ) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x minus 8 over 2 open parenthesis x plus 1 close parenthesis")[$frac(x − 8, 2 ( x + 1 ))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction y squared minus 6 y plus 8 over y squared minus 4 y divided by open parenthesis 3 y squared minus 12 y close parenthesis .")[$frac(y^(2) − 6 y + 8, y^(2) − 4 y) ÷ ( 3 y^(2) − 12 y ) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction y minus 2 over 3 y squared open parenthesis y minus 4 close parenthesis")[$frac(y − 2, 3 y^(2) ( y − 4 ))$]
]
]

Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

#examplebox("Example 12")[][
Divide: #math.equation(block: false, alt: "the fraction the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 over the fraction 2 x squared minus 7 x plus 3 over x squared minus 5 x plus 6 .")[$frac(frac(6 x^(2) − 7 x + 2, 4 x − 8), frac(2 x^(2) − 7 x + 3, x^(2) − 5 x + 6)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 over the fraction 2 x squared minus 7 x plus 3 over x squared minus 5 x plus 6")[$frac(frac(6 x^(2) − 7 x + 2, 4 x − 8), frac(2 x^(2) − 7 x + 3, x^(2) − 5 x + 6))$]]),
  [Rewrite with a division sign.], [#math.equation(block: false, alt: "the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 divided by the fraction 2 x squared minus 7 x plus 3 over x squared minus 5 x plus 6")[$frac(6 x^(2) − 7 x + 2, 4 x − 8) ÷ frac(2 x^(2) − 7 x + 3, x^(2) − 5 x + 6)$]],
  [Rewrite as product of first times #linebreak() reciprocal of second.], [#math.equation(block: false, alt: "the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 times the fraction x squared minus 5 x plus 6 over 2 x squared minus 7 x plus 3")[$frac(6 x^(2) − 7 x + 2, 4 x − 8) · frac(x^(2) − 5 x + 6, 2 x^(2) − 7 x + 3)$]],
  [Factor the numerators and the #linebreak() denominators, and then multiply.], [#math.equation(block: false, alt: "the fraction open parenthesis 2 x minus 1 close parenthesis open parenthesis 3 x minus 2 close parenthesis open parenthesis x minus 2 close parenthesis open parenthesis x minus 3 close parenthesis over 4 open parenthesis x minus 2 close parenthesis open parenthesis 2 x minus 1 close parenthesis open parenthesis x minus 3 close parenthesis")[$frac(( 2 x − 1 ) ( 3 x − 2 ) ( x − 2 ) ( x − 3 ), 4 ( x − 2 ) ( 2 x − 1 ) ( x − 3 ))$]],
  [Simplify by dividing out common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis 2 x minus 1 close parenthesis open parenthesis 3 x minus 2 close parenthesis open parenthesis x minus 2 close parenthesis open parenthesis x minus 3 close parenthesis over 4 open parenthesis x minus 2 close parenthesis open parenthesis 2 x minus 1 close parenthesis open parenthesis x minus 3 close parenthesis")[$frac(cancel(( 2 x − 1 )) ( 3 x − 2 ) cancel(( x − 2 )) cancel(( x − 3 )), 4 cancel(( x − 2 )) cancel(( 2 x − 1 )) cancel(( x − 3 )))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 x minus 2 over 4")[$frac(3 x − 2, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction the fraction 3 x squared plus 7 x plus 2 over 4 x plus 24 over the fraction 3 x squared minus 14 x minus 5 over x squared plus x minus 30 .")[$frac(frac(3 x^(2) + 7 x + 2, 4 x + 24), frac(3 x^(2) − 14 x − 5, x^(2) + x − 30)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x plus 2 over 4")[$frac(x + 2, 4)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction the fraction y squared minus 36 over 2 y squared plus 11 y minus 6 over the fraction 2 y squared minus 2 y minus 60 over 8 y minus 4 .")[$frac(frac(y^(2) − 36, 2 y^(2) + 11 y − 6), frac(2 y^(2) − 2 y − 60, 8 y − 4)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over y plus 5")[$frac(2, y + 5)$]
]
]

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.

#examplebox("Example 13")[][
Divide: #math.equation(block: false, alt: "the fraction 3 x minus 6 over 4 x minus 4 times the fraction x squared plus 2 x minus 3 over x squared minus 3 x minus 10 divided by the fraction 2 x plus 12 over 8 x plus 16 .")[$frac(3 x − 6, 4 x − 4) · frac(x^(2) + 2 x − 3, x^(2) − 3 x − 10) ÷ frac(2 x + 12, 8 x + 16) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the multiplication of (3x-6)/(4x-4) and (x^2+2x-3)/(x^2-3x-10), then divided by (2x+12)/(8x+16).], alt: "A mathematical expression showing the multiplication of (3x-6)/(4x-4) and (x^2+2x-3)/(x^2-3x-10), then divided by (2x+12)/(8x+16).", caption: none)]),
  [Rewrite the division as multiplication
by the reciprocal.], [#figure(figph[A mathematical expression showing the multiplication of three rational algebraic fractions: (3x-6)/(4x-4), (x^2+2x-3)/(x^2-3x-10), and (8x+16)/(2x+12), with the last fraction in red.], alt: "A mathematical expression showing the multiplication of three rational algebraic fractions: (3x-6)/(4x-4), (x^2+2x-3)/(x^2-3x-10), and (8x+16)/(2x+12), with the last fraction in red.", caption: none)],
  [Factor the numerators and the denominators,
and then multiply.], [#figure(figph[A complex algebraic fraction displays a numerator of 3 \* 8 \* (x-2)(x+3)(x-1)(x+2) and a denominator of 4 \* 2 \* (x-1)(x+2)(x-5)(x+6), set up for simplification.], alt: "A complex algebraic fraction displays a numerator of 3 * 8 * (x-2)(x+3)(x-1)(x+2) and a denominator of 4 * 2 * (x-1)(x+2)(x-5)(x+6), set up for simplification.", caption: none)],
  [Simplify by dividing out common factors.], [#figure(figph[A step-by-step simplification of a rational algebraic expression, demonstrating the cancellation of common factors such as (x-1), (x+2), and numerical terms in both the numerator and denominator.], alt: "A step-by-step simplification of a rational algebraic expression, demonstrating the cancellation of common factors such as (x-1), (x+2), and numerical terms in both the numerator and denominator.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing a fraction. The numerator is 3(x-2)(x+3) and the denominator is (x-5)(x+6).], alt: "A mathematical expression showing a fraction. The numerator is 3(x-2)(x+3) and the denominator is (x-5)(x+6).", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 4 m plus 4 over 3 m minus 15 times the fraction m squared minus 3 m minus 10 over m squared minus 4 m minus 32 divided by the fraction 12 m minus 36 over 6 m minus 48 .")[$frac(4 m + 4, 3 m − 15) · frac(m^(2) − 3 m − 10, m^(2) − 4 m − 32) ÷ frac(12 m − 36, 6 m − 48) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 open parenthesis m plus 1 close parenthesis open parenthesis m plus 2 close parenthesis over 3 open parenthesis m plus 4 close parenthesis open parenthesis m minus 3 close parenthesis")[$frac(2 ( m + 1 ) ( m + 2 ), 3 ( m + 4 ) ( m − 3 ))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 2 n squared plus 10 n over n minus 1 divided by the fraction n squared plus 10 n plus 24 over n squared plus 8 n minus 9 times the fraction n plus 4 over 8 n squared plus 12 n .")[$frac(2 n^(2) + 10 n, n − 1) ÷ frac(n^(2) + 10 n + 24, n^(2) + 8 n − 9) · frac(n + 4, 8 n^(2) + 12 n) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis n plus 5 close parenthesis open parenthesis n plus 9 close parenthesis over 2 open parenthesis n plus 6 close parenthesis open parenthesis 2 n plus 3 close parenthesis")[$frac(( n + 5 ) ( n + 9 ), 2 ( n + 6 ) ( 2 n + 3 ))$]
]
]


=== Key Concepts

- #strong[Multiplication of Rational Expressions]
- If #math.equation(block: false, alt: "p , q , r , s")[$p , q , r , s$] are polynomials where #math.equation(block: false, alt: "q not equal to 0 , s not equal to 0")[$q ≠ 0 , s ≠ 0$], then #math.equation(block: false, alt: "the fraction p over q times the fraction r over s equals the fraction p r over q s")[$frac(p, q) · frac(r, s) = frac(p r, q s)$].
- To multiply rational expressions, multiply the numerators and multiply the denominators
- #strong[Multiply a Rational Expression]
+ Factor each numerator and denominator completely.
+ Multiply the numerators and denominators.
+ Simplify by dividing out common factors.
- #strong[Division of Rational Expressions]
- If #math.equation(block: false, alt: "p , q , r , s")[$p , q , r , s$] are polynomials where #math.equation(block: false, alt: "q not equal to 0 , r not equal to 0 , s not equal to 0")[$q ≠ 0 , r ≠ 0 , s ≠ 0$], then #math.equation(block: false, alt: "the fraction p over q divided by the fraction r over s equals the fraction p over q times the fraction s over r")[$frac(p, q) ÷ frac(r, s) = frac(p, q) · frac(s, r)$].
- To divide rational expressions multiply the first fraction by the reciprocal of the second.
- #strong[Divide Rational Expressions]
+ Rewrite the division as the product of the first rational expression and the reciprocal of the second.
+ Factor the numerators and denominators completely.
+ Multiply the numerators and denominators together. #linebreak()  #linebreak()
+ Simplify by dividing out common factors.


==== Practice Makes Perfect

#strong[Multiply Rational Expressions]

In the following exercises, multiply.

#math.equation(block: true, alt: "the fraction 12 over 16 times the fraction 4 over 10")[$frac(12, 16) · frac(4, 10)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 10")[$frac(3, 10)$]
]

#math.equation(block: true, alt: "the fraction 32 over 5 times the fraction 16 over 24")[$frac(32, 5) · frac(16, 24)$]

#math.equation(block: true, alt: "the fraction 18 over 10 times the fraction 4 over 30")[$frac(18, 10) · frac(4, 30)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 over 25")[$frac(6, 25)$]
]

#math.equation(block: true, alt: "the fraction 21 over 36 times the fraction 45 over 24")[$frac(21, 36) · frac(45, 24)$]

#math.equation(block: true, alt: "the fraction 5 x squared y to the power 4 over 12 x y cubed times the fraction 6 x squared over 20 y squared")[$frac(5 x^(2) y^(4), 12 x y^(3)) · frac(6 x^(2), 20 y^(2))$]

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

#math.equation(block: true, alt: "the fraction 8 w cubed y over 9 y squared times the fraction 3 y over 4 w to the power 4")[$frac(8 w^(3) y, 9 y^(2)) · frac(3 y, 4 w^(4))$]

#math.equation(block: true, alt: "the fraction 12 a cubed b over b squared times the fraction 2 a b squared over 9 b cubed")[$frac(12 a^(3) b, b^(2)) · frac(2 a b^(2), 9 b^(3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 a to the power 4 over 3 b squared")[$frac(8 a^(4), 3 b^(2))$]
]

#math.equation(block: true, alt: "the fraction 4 m n squared over 5 n cubed times the fraction m n cubed over 8 m squared n squared")[$frac(4 m n^(2), 5 n^(3)) · frac(m n^(3), 8 m^(2) n^(2))$]

#math.equation(block: true, alt: "the fraction 5 p squared over p squared minus 5 p minus 36 times the fraction p squared minus 16 over 10 p")[$frac(5 p^(2), p^(2) − 5 p − 36) · frac(p^(2) − 16, 10 p)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction p open parenthesis p minus 4 close parenthesis over 2 open parenthesis p minus 9 close parenthesis")[$frac(p ( p − 4 ), 2 ( p − 9 ))$]
]

#math.equation(block: true, alt: "the fraction 3 q squared over q squared plus q minus 6 times the fraction q squared minus 9 over 9 q")[$frac(3 q^(2), q^(2) + q − 6) · frac(q^(2) − 9, 9 q)$]

#math.equation(block: true, alt: "the fraction 4 r over r squared minus 3 r minus 10 times the fraction r squared minus 25 over 8 r squared")[$frac(4 r, r^(2) − 3 r − 10) · frac(r^(2) − 25, 8 r^(2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction r plus 5 over 2 r open parenthesis r plus 2 close parenthesis")[$frac(r + 5, 2 r ( r + 2 ))$]
]

#math.equation(block: true, alt: "the fraction s over s squared minus 9 s plus 14 times the fraction s squared minus 49 over 7 s squared")[$frac(s, s^(2) − 9 s + 14) · frac(s^(2) − 49, 7 s^(2))$]

#math.equation(block: true, alt: "the fraction x squared minus 7 x over x squared plus 6 x plus 9 times the fraction x plus 3 over 4 x")[$frac(x^(2) − 7 x, x^(2) + 6 x + 9) · frac(x + 3, 4 x)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x minus 7 over 4 open parenthesis x plus 3 close parenthesis")[$frac(x − 7, 4 ( x + 3 ))$]
]

#math.equation(block: true, alt: "the fraction 2 y squared minus 10 y over y squared plus 10 y plus 25 times the fraction y plus 5 over 6 y")[$frac(2 y^(2) − 10 y, y^(2) + 10 y + 25) · frac(y + 5, 6 y)$]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction z plus 3 over z open parenthesis z plus 1 close parenthesis")[$frac(z + 3, z ( z + 1 ))$]
]

#math.equation(block: true, alt: "the fraction 2 a squared plus 8 a over a squared minus 9 a plus 20 times the fraction a minus 5 over a squared")[$frac(2 a^(2) + 8 a, a^(2) − 9 a + 20) · frac(a − 5, a^(2))$]

#math.equation(block: true, alt: "the fraction 28 minus 4 b over 3 b minus 3 times the fraction b squared plus 8 b minus 9 over b squared minus 49")[$frac(28 − 4 b, 3 b − 3) · frac(b^(2) + 8 b − 9, b^(2) − 49)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 4 open parenthesis b plus 9 close parenthesis over 3 open parenthesis b plus 7 close parenthesis")[$− frac(4 ( b + 9 ), 3 ( b + 7 ))$]
]

#math.equation(block: true, alt: "the fraction 18 c minus 2 c squared over 6 c plus 30 times the fraction c squared plus 7 c plus 10 over c squared minus 81")[$frac(18 c − 2 c^(2), 6 c + 30) · frac(c^(2) + 7 c + 10, c^(2) − 81)$]

#math.equation(block: true, alt: "the fraction 35 d minus 7 d squared over d squared plus 7 d times the fraction d squared plus 12 d plus 35 over d squared minus 25")[$frac(35 d − 7 d^(2), d^(2) + 7 d) · frac(d^(2) + 12 d + 35, d^(2) − 25)$]

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

#math.equation(block: true, alt: "the fraction 72 m minus 12 m squared over 8 m plus 32 times the fraction m squared plus 10 m plus 24 over m squared minus 36")[$frac(72 m − 12 m^(2), 8 m + 32) · frac(m^(2) + 10 m + 24, m^(2) − 36)$]

#math.equation(block: true, alt: "the fraction 4 n plus 20 over n squared plus n minus 20 times the fraction n squared minus 16 over 4 n plus 16")[$frac(4 n + 20, n^(2) + n − 20) · frac(n^(2) − 16, 4 n + 16)$]

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

#math.equation(block: true, alt: "the fraction 6 p squared minus 6 p over p squared plus 7 p minus 18 times the fraction p squared minus 81 over 3 p squared minus 27 p")[$frac(6 p^(2) − 6 p, p^(2) + 7 p − 18) · frac(p^(2) − 81, 3 p^(2) − 27 p)$]

#math.equation(block: true, alt: "the fraction q squared minus 2 q over q squared plus 6 q minus 16 times the fraction q squared minus 64 over q squared minus 8 q")[$frac(q^(2) − 2 q, q^(2) + 6 q − 16) · frac(q^(2) − 64, q^(2) − 8 q)$]

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

#math.equation(block: true, alt: "the fraction 2 r squared minus 2 r over r squared plus 4 r minus 5 times the fraction r squared minus 25 over 2 r squared minus 10 r")[$frac(2 r^(2) − 2 r, r^(2) + 4 r − 5) · frac(r^(2) − 25, 2 r^(2) − 10 r)$]

#strong[Divide Rational Expressions]

In the following exercises, divide.

#math.equation(block: true, alt: "the fraction t minus 6 over 3 minus t divided by the fraction t minus 5 over t squared minus 9")[$frac(t − 6, 3 − t) ÷ frac(t − 5, t^(2) − 9)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 minus t t plus 3 over t minus 5")[$frac(6 − t t + 3, t − 5)$]
]

#math.equation(block: true, alt: "the fraction v minus 5 over 11 minus v divided by the fraction v squared minus 25 over v minus 11")[$frac(v − 5, 11 − v) ÷ frac(v^(2) − 25, v − 11)$]

#math.equation(block: true, alt: "the fraction 10 plus w over w minus 8 divided by the fraction 100 minus w squared over 8 minus w")[$frac(10 + w, w − 8) ÷ frac(100 − w^(2), 8 − w)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 10 minus w")[$− frac(1, 10 − w)$]
]

#math.equation(block: true, alt: "the fraction 7 plus x over x minus 6 divided by the fraction 49 minus x over x plus 6 squared")[$frac(7 + x, x − 6) ÷ attach(frac(49 − x, x + 6), t: 2)$]

#math.equation(block: true, alt: "the fraction 27 y squared over 3 y minus 21 divided by the fraction 3 y squared plus 18 over y squared plus 13 y plus 42")[$frac(27 y^(2), 3 y − 21) ÷ frac(3 y^(2) + 18, y^(2) + 13 y + 42)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 y squared open parenthesis y plus 6 close parenthesis open parenthesis y plus 7 close parenthesis over open parenthesis y minus 7 close parenthesis open parenthesis y squared plus 6 close parenthesis")[$frac(3 y^(2) ( y + 6 ) ( y + 7 ), ( y − 7 ) ( y^(2) + 6 ))$]
]

#math.equation(block: true, alt: "the fraction 24 z squared over 2 z minus 8 divided by the fraction 4 z minus 28 over z squared minus 11 z plus 28")[$frac(24 z^(2), 2 z − 8) ÷ frac(4 z − 28, z^(2) − 11 z + 28)$]

#math.equation(block: true, alt: "the fraction 16 a squared over 4 a plus 36 divided by the fraction 4 a squared minus 24 a over a squared plus 4 a minus 45")[$frac(16 a^(2), 4 a + 36) ÷ frac(4 a^(2) − 24 a, a^(2) + 4 a − 45)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction a open parenthesis a minus 5 close parenthesis over a minus 6")[$frac(a ( a − 5 ), a − 6)$]
]

#math.equation(block: true, alt: "the fraction 24 b squared over 2 b minus 4 divided by the fraction 12 b squared plus 36 b over b squared minus 11 b plus 18")[$frac(24 b^(2), 2 b − 4) ÷ frac(12 b^(2) + 36 b, b^(2) − 11 b + 18)$]

#math.equation(block: true, alt: "the fraction 5 c squared plus 9 c minus 2 over c squared minus 4 divided by the fraction 5 c squared minus 16 c plus 3 over c squared plus 4 c plus 4")[$frac(5 c^(2) + 9 c − 2, c^(2) − 4) ÷ frac(5 c^(2) − 16 c + 3, c^(2) + 4 c + 4)$]

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

#math.equation(block: true, alt: "the fraction 2 d squared plus d minus 3 over d squared minus 16 divided by the fraction 2 d squared minus 9 d minus 18 over d squared minus 8 d plus 16")[$frac(2 d^(2) + d − 3, d^(2) − 16) ÷ frac(2 d^(2) − 9 d − 18, d^(2) − 8 d + 16)$]

#math.equation(block: true, alt: "the fraction 6 m squared minus 11 m minus 2 over 9 minus m squared divided by the fraction 6 m squared plus 25 m plus 4 over m squared minus 6 m plus 9")[$frac(6 m^(2) − 11 m − 2, 9 − m^(2)) ÷ frac(6 m^(2) + 25 m + 4, m^(2) − 6 m + 9)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction open parenthesis m minus 2 close parenthesis open parenthesis m minus 3 close parenthesis over open parenthesis 3 plus m close parenthesis open parenthesis m plus 4 close parenthesis")[$− frac(( m − 2 ) ( m − 3 ), ( 3 + m ) ( m + 4 ))$]
]

#math.equation(block: true, alt: "the fraction 2 n squared minus 3 n minus 14 over 25 minus n squared divided by the fraction 2 n squared minus 13 n plus 21 over n squared minus 10 n plus 25")[$frac(2 n^(2) − 3 n − 14, 25 − n^(2)) ÷ frac(2 n^(2) − 13 n + 21, n^(2) − 10 n + 25)$]

#math.equation(block: true, alt: "the fraction 3 s squared over s squared minus 16 divided by the fraction s cubed plus 4 s squared plus 16 s over s cubed minus 64")[$frac(3 s^(2), s^(2) − 16) ÷ frac(s^(3) + 4 s^(2) + 16 s, s^(3) − 64)$]

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

#math.equation(block: true, alt: "the fraction r squared minus 9 over 15 divided by the fraction r cubed minus 27 over 5 r squared plus 15 r plus 45")[$frac(r^(2) − 9, 15) ÷ frac(r^(3) − 27, 5 r^(2) + 15 r + 45)$]

#math.equation(block: true, alt: "the fraction p cubed plus q cubed over 3 p squared plus 3 p q plus 3 q squared divided by the fraction p squared minus q squared over 12")[$frac(p^(3) + q^(3), 3 p^(2) + 3 p q + 3 q^(2)) ÷ frac(p^(2) − q^(2), 12)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 open parenthesis p squared minus p q plus q squared close parenthesis over open parenthesis p minus q close parenthesis open parenthesis p squared plus p q plus q squared close parenthesis")[$frac(4 ( p^(2) − p q + q^(2) ), ( p − q ) ( p^(2) + p q + q^(2) ))$]
]

#math.equation(block: true, alt: "the fraction v cubed minus 8 w cubed over 2 v squared plus 4 v w plus 8 w squared divided by the fraction v squared minus 4 w squared over 4")[$frac(v^(3) − 8 w^(3), 2 v^(2) + 4 v w + 8 w^(2)) ÷ frac(v^(2) − 4 w^(2), 4)$]

#math.equation(block: true, alt: "the fraction t squared minus 9 over 2 t divided by open parenthesis t squared minus 6 t plus 9 close parenthesis")[$frac(t^(2) − 9, 2 t) ÷ ( t^(2) − 6 t + 9 )$]

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

#math.equation(block: true, alt: "the fraction x squared plus 3 x minus 10 over 4 x divided by open parenthesis 2 x squared plus 20 x plus 50 close parenthesis")[$frac(x^(2) + 3 x − 10, 4 x) ÷ ( 2 x^(2) + 20 x + 50 )$]

#math.equation(block: true, alt: "the fraction 2 y squared minus 10 y z minus 48 z squared over 2 y minus 1 divided by open parenthesis 4 y squared minus 32 y z close parenthesis")[$frac(2 y^(2) − 10 y z − 48 z^(2), 2 y − 1) ÷ ( 4 y^(2) − 32 y z )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction y plus 3 z over 2 y open parenthesis 2 y minus 1 close parenthesis")[$frac(y + 3 z, 2 y ( 2 y − 1 ))$]
]

#math.equation(block: true, alt: "the fraction 2 m squared minus 98 n squared over 2 m plus 6 divided by open parenthesis m squared minus 7 m n close parenthesis")[$frac(2 m^(2) − 98 n^(2), 2 m + 6) ÷ ( m^(2) − 7 m n )$]

#math.equation(block: true, alt: "the fraction the fraction 2 a squared minus a minus 21 over 5 a plus 20 over the fraction a squared plus 7 a plus 12 over a squared plus 8 a plus 16")[$frac(frac(2 a^(2) − a − 21, 5 a + 20), frac(a^(2) + 7 a + 12, a^(2) + 8 a + 16))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 a minus 7 over 5")[$frac(2 a − 7, 5)$]
]

#math.equation(block: true, alt: "the fraction the fraction 3 b squared plus 2 b minus 8 over 12 b plus 18 over the fraction 3 b squared plus 2 b minus 8 over 2 b squared minus 7 b minus 15")[$frac(frac(3 b^(2) + 2 b − 8, 12 b + 18), frac(3 b^(2) + 2 b − 8, 2 b^(2) − 7 b − 15))$]

#math.equation(block: true, alt: "the fraction the fraction 12 c squared minus 12 over 2 c squared minus 3 c plus 1 over the fraction 4 c plus 4 over 6 c squared minus 13 c plus 5")[$frac(frac(12 c^(2) − 12, 2 c^(2) − 3 c + 1), frac(4 c + 4, 6 c^(2) − 13 c + 5))$]

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

#math.equation(block: true, alt: "the fraction the fraction 4 d squared plus 7 d minus 2 over 35 d plus 10 over the fraction d squared minus 4 over 7 d squared minus 12 d minus 4")[$frac(frac(4 d^(2) + 7 d − 2, 35 d + 10), frac(d^(2) − 4, 7 d^(2) − 12 d − 4))$]

#math.equation(block: true, alt: "the fraction 10 m squared plus 80 m over 3 m minus 9 times the fraction m squared plus 4 m minus 21 over m squared minus 9 m plus 20")[$frac(10 m^(2) + 80 m, 3 m − 9) · frac(m^(2) + 4 m − 21, m^(2) − 9 m + 20)$] #linebreak() #math.equation(block: true, alt: "divided by the fraction 5 m squared plus 10 m over 2 m minus 10")[$#h(1.5em) ÷ frac(5 m^(2) + 10 m, 2 m − 10)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 open parenthesis m plus 8 close parenthesis open parenthesis m plus 7 close parenthesis over 3 open parenthesis m minus 4 close parenthesis open parenthesis m plus 2 close parenthesis")[$frac(4 ( m + 8 ) ( m + 7 ), 3 ( m − 4 ) ( m + 2 ))$]
]

#math.equation(block: true, alt: "the fraction 4 n squared plus 32 n over 3 n plus 2 times the fraction 3 n squared minus n minus 2 over n squared plus n minus 30")[$frac(4 n^(2) + 32 n, 3 n + 2) · frac(3 n^(2) − n − 2, n^(2) + n − 30)$] #linebreak() #math.equation(block: true, alt: "divided by the fraction 108 n squared minus 24 n over n plus 6")[$#h(1.5em) ÷ frac(108 n^(2) − 24 n, n + 6)$]

#math.equation(block: true, alt: "the fraction 12 p squared plus 3 p over p plus 3 divided by the fraction p squared plus 2 p minus 63 over p squared minus p minus 12")[$frac(12 p^(2) + 3 p, p + 3) ÷ frac(p^(2) + 2 p − 63, p^(2) − p − 12)$] #linebreak() #math.equation(block: true, alt: "times the fraction p minus 7 over 9 p cubed minus 9 p squared")[$#h(1.5em) · frac(p − 7, 9 p^(3) − 9 p^(2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis 4 p plus 1 close parenthesis open parenthesis p minus 4 close parenthesis over 3 p open parenthesis p plus 9 close parenthesis open parenthesis p minus 1 close parenthesis")[$frac(( 4 p + 1 ) ( p − 4 ), 3 p ( p + 9 ) ( p − 1 ))$]
]

#math.equation(block: true, alt: "the fraction 6 q plus 3 over 9 q squared minus 9 q divided by the fraction q squared plus 14 q plus 33 over q squared plus 4 q minus 5")[$frac(6 q + 3, 9 q^(2) − 9 q) ÷ frac(q^(2) + 14 q + 33, q^(2) + 4 q − 5)$] #linebreak() #math.equation(block: true, alt: "times the fraction 4 q squared plus 12 q over 12 q plus 6")[$#h(1.5em) · frac(4 q^(2) + 12 q, 12 q + 6)$]


==== Everyday Math

#strong[Probability] The director of large company is interviewing applicants for two identical jobs. If #math.equation(block: false, alt: "w equals")[$w =$] the number of women applicants and #math.equation(block: false, alt: "m equals")[$m =$] the number of men applicants, then the probability that two women are selected for the jobs is #math.equation(block: false, alt: "the fraction w over w plus m times the fraction w minus 1 over w plus m minus 1 .")[$frac(w, w + m) · frac(w − 1, w + m − 1) .$]

+ ⓐ Simplify the probability by multiplying the two rational expressions.
+ ⓑ Find the probability that two women are selected when #math.equation(block: false, alt: "w equals 5")[$w = 5$] and #math.equation(block: false, alt: "m equals 10")[$m = 10$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction w open parenthesis w minus 1 close parenthesis over open parenthesis w plus m close parenthesis open parenthesis w plus m minus 1 close parenthesis")[$frac(w ( w − 1 ), ( w + m ) ( w + m − 1 ))$] #linebreak() ⓑ #math.equation(block: false, alt: "the fraction 2 over 21")[$frac(2, 21)$]
]

#strong[Area of a triangle] The area of a triangle with base b and height h is #math.equation(block: false, alt: "the fraction b h over 2 .")[$frac(b h, 2) .$] If the triangle is stretched to make a new triangle with base and height three times as much as in the original triangle, the area is #math.equation(block: false, alt: "the fraction 9 b h over 2 .")[$frac(9 b h, 2) .$] Calculate how the area of the new triangle compares to the area of the original triangle by dividing #math.equation(block: false, alt: "the fraction 9 b h over 2")[$frac(9 b h, 2)$] by #math.equation(block: false, alt: "the fraction b h over 2")[$frac(b h, 2)$].


==== Writing Exercises

+ ⓐ Multiply #math.equation(block: false, alt: "the fraction 7 over 4 times the fraction 9 over 10")[$frac(7, 4) · frac(9, 10)$] and explain all your steps.
+ ⓑ Multiply #math.equation(block: false, alt: "the fraction n over n minus 3 times the fraction 9 over n plus 3")[$frac(n, n − 3) · frac(9, n + 3)$] and explain all your steps.
+ ⓒ Evaluate your answer to part (b) when #math.equation(block: false, alt: "n equals 7 .")[$n = 7 .$] Did you get the same answer you got in part (a)? Why or why not?

#solutionbox[
Answers will vary.
]

+ ⓐ Divide #math.equation(block: false, alt: "the fraction 24 over 5 divided by 6")[$frac(24, 5) ÷ 6$] and explain all your steps.
+ ⓑ Divide #math.equation(block: false, alt: "the fraction x squared minus 1 over x divided by open parenthesis x plus 1 close parenthesis")[$frac(x^(2) − 1, x) ÷ ( x + 1 )$] and explain all your steps.
+ ⓒ Evaluate your answer to part (b) when #math.equation(block: false, alt: "x equals 5 .")[$x = 5 .$] Did you get the same answer you got in part (a)? Why or why not?


==== Self Check

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

#figure(figph[The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads multiply rational expressions.”, the next row reads “divide rational expressions.”, the last row reads “after reviewing this checklist, what will you do to become confident for all objectives?” The remaining columns are blank.], alt: "The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads multiply rational expressions.”, the next row reads “divide rational expressions.”, the last row reads “after reviewing this checklist, what will you do to become confident for all objectives?” The remaining columns are blank.", caption: none)

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?