Question

perform the following operation and express in simplest form.\\(\dfrac{x^2 - 81}{x - 9} \div \dfrac{x^2 - 17x + 72}{x - 8}\\)

Explanation:

Step1: Rewrite division as multiplication

$\frac{x^2 - 81}{x - 9} \times \frac{x - 8}{x^2 - 17x + 72}$

Step2: Factor all polynomials

$x^2-81=(x-9)(x+9)$; $x^2-17x+72=(x-8)(x-9)$
Substitute: $\frac{(x-9)(x+9)}{x - 9} \times \frac{x - 8}{(x-8)(x-9)}$

Step3: Cancel common factors

Cancel $(x-9)$ (numerator/denominator 1), cancel $(x-8)$:
$\frac{x+9}{1} \times \frac{1}{x-9}$

Step4: Multiply remaining terms

$\frac{(x+9) \times 1}{1 \times (x-9)}$

Answer:

$\frac{x+9}{x-9}$