Question

example 5
simplify each expression.

1. (dfrac{dfrac{x^2 - 9}{6x - 12}}{dfrac{x^2 + 10x + 21}{x^2 - x - 2}})
2. (dfrac{dfrac{y - x}{z^3}}{dfrac{x - y}{6z^2}})
3. (dfrac{dfrac{a^2 - b^2}{b^3}}{dfrac{b^2 - ab}{a^2}})
4. (dfrac{dfrac{x - y}{a + b}}{dfrac{x^2 - y^2}{b^2 - a^2}})

mixed exercises
simplify each expression.

1. (dfrac{dfrac{y^2 + 8y + 15}{y - 6}}{dfrac{y^2 - 9y + 18}{y^2 - 9}})
2. (dfrac{dfrac{c^2 - 6c - 16}{c^2 - a^2}}{dfrac{c^2 - 8c}{c + a}})

Explanation:

Problem 20

Step1: Rewrite as multiplication

$\frac{x^2 - 9}{6x - 12} \times \frac{x^2 - x - 2}{x^2 + 10x + 21}$

Step2: Factor all polynomials

$\frac{(x-3)(x+3)}{6(x-2)} \times \frac{(x-2)(x+1)}{(x+3)(x+7)}$

Step3: Cancel common factors

$\frac{(x-3)\cancel{(x+3)}}{6\cancel{(x-2)}} \times \frac{\cancel{(x-2)}(x+1)}{\cancel{(x+3)}(x+7)}$

Step4: Multiply remaining terms

$\frac{(x-3)(x+1)}{6(x+7)}$

Problem 21

Step1: Rewrite as multiplication

$\frac{y - x}{z^3} \times \frac{6z^2}{x - y}$

Step2: Rewrite $y-x$ as $-(x-y)$

$\frac{-(x-y)}{z^3} \times \frac{6z^2}{x - y}$

Step3: Cancel common factors

$\frac{-\cancel{(x-y)}}{\cancel{z^2} \cdot z} \times \frac{6\cancel{z^2}}{\cancel{(x-y)}}$

Step4: Simplify the expression

$-\frac{6}{z}$

Problem 22

Step1: Rewrite as multiplication

$\frac{a^2 - b^2}{b^3} \times \frac{a^2}{b^2 - ab}$

Step2: Factor all polynomials

$\frac{(a-b)(a+b)}{b^3} \times \frac{a^2}{b(b - a)}$

Step3: Rewrite $b-a$ as $-(a-b)$

$\frac{(a-b)(a+b)}{b^3} \times \frac{a^2}{-b(a - b)}$

Step4: Cancel common factors

$\frac{\cancel{(a-b)}(a+b)}{b^3} \times \frac{a^2}{-b\cancel{(a - b)}}$

Step5: Multiply remaining terms

$-\frac{a^2(a+b)}{b^4}$

Problem 23

Step1: Rewrite as multiplication

$\frac{x - y}{\frac{a+b}{x^2 - y^2}} \times \frac{1}{b^2 - a^2}$
$\frac{x - y}{1} \times \frac{x^2 - y^2}{a+b} \times \frac{1}{b^2 - a^2}$

Step2: Factor all polynomials

$\frac{x - y}{1} \times \frac{(x-y)(x+y)}{a+b} \times \frac{1}{-(a^2 - b^2)}$
$\frac{x - y}{1} \times \frac{(x-y)(x+y)}{a+b} \times \frac{1}{-(a-b)(a+b)}$

Step3: Cancel common factors

$\frac{\cancel{x - y}}{1} \times \frac{(\cancel{x-y})(x+y)}{\cancel{a+b}} \times \frac{1}{-(\cancel{a-b})(a+b)}$

Step4: Simplify the expression

$-\frac{x+y}{a+b}$

Problem 24

Step1: Factor all polynomials

$\frac{(y+3)(y+5)}{y-6} \times \frac{(y-3)(y-6)}{(y-3)(y+3)}$

Step2: Cancel common factors

$\frac{\cancel{(y+3)}(y+5)}{\cancel{y-6}} \times \frac{\cancel{(y-3)}\cancel{(y-6)}}{\cancel{(y-3)}\cancel{(y+3)}}$

Step3: Simplify the product

$y+5$

Problem 25

Step1: Rewrite as multiplication

$\frac{c^2 - 6c - 16}{c^2 - d^2} \times \frac{c + d}{c^2 - 8c}$

Step2: Factor all polynomials

$\frac{(c-8)(c+2)}{(c-d)(c+d)} \times \frac{c + d}{c(c-8)}$

Step3: Cancel common factors

$\frac{\cancel{(c-8)}(c+2)}{(c-d)\cancel{(c+d)}} \times \frac{\cancel{c + d}}{c\cancel{(c-8)}}$

Step4: Simplify the product

$\frac{c+2}{c(c-d)}$

Answer:

1. $\boldsymbol{\frac{(x-3)(x+1)}{6(x+7)}}$
2. $\boldsymbol{-\frac{6}{z}}$
3. $\boldsymbol{-\frac{a^2(a+b)}{b^4}}$
4. $\boldsymbol{-\frac{x+y}{a+b}}$
5. $\boldsymbol{y+5}$
6. $\boldsymbol{\frac{c+2}{c(c-d)}}$