Question

write the rational expression in lowest terms.\\(
\frac{x^{2}-5x + 4}{x^{2}+x - 20}\\)\\(
\frac{x^{2}-5x + 4}{x^{2}+x - 20}=\square\\)\\(\text{(simplify your answer.)}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 - 5x + 4\): We need two numbers that multiply to \(4\) and add to \(-5\). Those numbers are \(-1\) and \(-4\), so \(x^2 - 5x + 4=(x - 1)(x - 4)\).

Factor \(x^2 + x - 20\): We need two numbers that multiply to \(-20\) and add to \(1\). Those numbers are \(5\) and \(-4\), so \(x^2 + x - 20=(x + 5)(x - 4)\).

So the rational expression becomes \(\frac{(x - 1)(x - 4)}{(x + 5)(x - 4)}\).

Step2: Cancel common factors

The common factor in the numerator and denominator is \((x - 4)\) (assuming \(x
eq4\) to avoid division by zero). Canceling this factor, we get \(\frac{x - 1}{x + 5}\).

Answer:

\(\frac{x - 1}{x + 5}\)