Question

simplifying rational expressions
simplify the rational expression.
\\(\frac{x^2 - x - 6}{x^2 - 3x - 10}\\)
\\(\frac{x - 3}{x - 5}\\) \\(\frac{x + 2}{x - 5}\\) \\(\frac{x - 3}{x + 2}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 - x - 6\): find two numbers that multiply to \(-6\) and add to \(-1\), which are \(-3\) and \(2\). So \(x^2 - x - 6=(x - 3)(x + 2)\).
Factor \(x^2 - 3x - 10\): find two numbers that multiply to \(-10\) and add to \(-3\), which are \(-5\) and \(2\). So \(x^2 - 3x - 10=(x - 5)(x + 2)\).

Step2: Simplify the rational expression

Substitute the factored forms into the expression: \(\frac{(x - 3)(x + 2)}{(x - 5)(x + 2)}\). Cancel out the common factor \((x + 2)\) (assuming \(x
eq - 2\) to avoid division by zero), resulting in \(\frac{x - 3}{x - 5}\).

Answer:

\(\boldsymbol{\frac{x - 3}{x - 5}}\) (corresponding to the first option box)