Chapter P - Section P.6 - Rational Expressions - Exercise Set - Page 84: 26

$=4(x-2),\qquad x\neq-2,2$

Work Step by Step

Factor what we can. Recognize a difference of squares. $\displaystyle \frac{x^{2}-4}{x-2}\div\frac{x+2}{4x-8}=\frac{(x-2)(x+2)}{x-2}\div\frac{x+2}{4(x-2)}$ Division with $\displaystyle \frac{a}{b}$ equals multiplication with $\displaystyle \frac{b}{a}.$ (neither a or b can be 0, we exclude $-2$ and $2$) $=\displaystyle \frac{(x-2)(x+2)}{x-2}\cdot\frac{4(x-2)}{x+2}$ ...exclude any additional values that yield 0 in the denominator: $x\neq-2,2$ ... cancel common factors $=\displaystyle \frac{4(x-2)}{1},\qquad x\neq-2,2$ $=4(x-2),\qquad x\neq-2,2$

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