Question

perform the following operation and express in simplest form.
$\frac{x^{2}-11x + 30}{x - 6} div \frac{x^{2}-12x + 35}{9x^{3}}$

Explanation:

Step1: Rewrite division as multiplication

$\frac{x^2 - 11x + 30}{x - 6} \times \frac{9x^3}{x^2 - 12x + 35}$

Step2: Factor quadratic numerators

Factor $x^2-11x+30$: find two numbers that multiply to 30 and add to -11, which are -5 and -6. So $x^2-11x+30=(x-5)(x-6)$.
Factor $x^2-12x+35$: find two numbers that multiply to 35 and add to -12, which are -5 and -7. So $x^2-12x+35=(x-5)(x-7)$.
Substitute back:
$\frac{(x-5)(x-6)}{x - 6} \times \frac{9x^3}{(x-5)(x-7)}$

Step3: Cancel common factors

Cancel $(x-6)$ from numerator and denominator:
$\frac{(x-5)}{1} \times \frac{9x^3}{(x-5)(x-7)}$
Cancel $(x-5)$ from numerator and denominator:
$\frac{1}{1} \times \frac{9x^3}{x-7}$

Step4: Simplify the product

Multiply the remaining terms:
$\frac{9x^3}{x-7}$

Answer:

$\frac{9x^3}{x-7}$ (where $x
eq 5, 6, 7, 0$)