Question

question 31
factor completely. if the polynomial is prime, say so.
$5x^2 - 45x + 100$
\\(\bigcirc\\) $5(x - 20)(x + 1)$
\\(\bigcirc\\) $(5x - 20)(x - 5)$
\\(\bigcirc\\) prime
\\(\bigcirc\\) $5(x - 4)(x - 5)$

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(5x^2\), \(-45x\), and \(100\) is \(5\). Factor out \(5\):
\(5x^2 - 45x + 100 = 5(x^2 - 9x + 20)\)

Step2: Factor the quadratic

Factor \(x^2 - 9x + 20\). We need two numbers that multiply to \(20\) and add to \(-9\). These numbers are \(-4\) and \(-5\):
\(x^2 - 9x + 20 = (x - 4)(x - 5)\)

Step3: Combine results

Substitute the factored quadratic back into the expression:
\(5(x^2 - 9x + 20) = 5(x - 4)(x - 5)\)

Answer:

\(5(x - 4)(x - 5)\) (corresponding to the option "5(x - 4)(x - 5)")