Question

which expression shows \\(\frac{x^3 - 12x^2 + 35x}{x^3 - 125}\\) in simplest form?\
\\(\bigcirc\\) \\(\frac{-7x}{5x + 25}\\)\
\\(\bigcirc\\) \\(\frac{x^2 - 7x}{x^2 + 5x + 25}\\)\
\\(\bigcirc\\) \\(\frac{x - 7}{x^2 + 5x + 25}\\)\
\\(\bigcirc\\) \\(\frac{x^2 - 7x}{x^2 + 10x + 25}\\)

Explanation:

Step1: Factor numerator

First, factor out $x$ from the numerator, then factor the quadratic:
$x^3 - 12x^2 + 35x = x(x^2 - 12x + 35) = x(x-5)(x-7)$

Step2: Factor denominator

Use difference of cubes formula $a^3-b^3=(a-b)(a^2+ab+b^2)$:
$x^3 - 125 = x^3 - 5^3 = (x-5)(x^2+5x+25)$

Step3: Cancel common factors

Cancel the $(x-5)$ term (where $x
eq5$):
$\frac{x(x-5)(x-7)}{(x-5)(x^2+5x+25)} = \frac{x(x-7)}{x^2+5x+25}$

Step4: Simplify numerator

Expand the remaining numerator:
$\frac{x(x-7)}{x^2+5x+25} = \frac{x^2-7x}{x^2+5x+25}$

Answer:

$\boldsymbol{\frac{x^2 - 7x}{x^2 + 5x + 25}}$ (corresponding to the second option)