Question

example 5 simplify each expression. 20. \\(\dfrac{\dfrac{\dfrac{x^2 - 9}{6x - 12}}{x^2 + 10x + 21}}{x^2 - x - 2}\\) 21. \\(\dfrac{\dfrac{y - x}{z^3}}{\dfrac{x - y}{6z^2}}\\) 22. \\(\dfrac{\dfrac{\dfrac{a^2 - b^2}{b^3}}{b^2 - ab}}{a^2}\\) 23. \\(\dfrac{\dfrac{\dfrac{x - y}{a + b}}{x^2 - y^2}}{b^2 - a^2}\\)

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: Factor out -1 from $y-x$

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

Step3: Cancel common factors

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

Step4: Simplify remaining terms

$-\frac{6}{z}$

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: Factor out -1 from $b-a$

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

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

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

Step2: Factor all polynomials

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

Step3: Cancel common factors

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

Step4: Multiply remaining terms

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

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. $\boldsymbol{-\frac{(x-y)^2(x+y)}{(a-b)(a+b)}}$