Question

select the correct answer. assuming that no denominator equals zero, what is the simplest form of this expression? \\(\frac{x - 1}{x^2 - 4x + 3} div \frac{5x - 2}{x^2 - 9}\\) \\(\bigcirc) \\(\frac{x - 1}{x + 3}\\) \\(\bigcirc) \\(\frac{x + 3}{5x - 2}\\) \\(\bigcirc) \\(\frac{3}{5x - 2}\\) \\(\bigcirc) \\(\frac{1}{5x - 2}\\)

Explanation:

Step1: Rewrite division as multiplication

$\frac{x-1}{x^2-4x+3} \times \frac{x^2-9}{5x-2}$

Step2: Factor all polynomials

$\frac{x-1}{(x-1)(x-3)} \times \frac{(x-3)(x+3)}{5x-2}$

Step3: Cancel common factors

$\frac{\cancel{x-1}}{\cancel{(x-1)}\cancel{(x-3)}} \times \frac{\cancel{(x-3)}(x+3)}{5x-2}$

Step4: Multiply remaining terms

$\frac{x+3}{5x-2}$

Answer:

$\boldsymbol{\frac{x+3}{5x-2}}$ (matches the second option)