Question

if $x - 9$ is a factor of $x^2 - 5x - 36$, what is the other factor?
$\bigcirc$ $x - 4$
$\bigcirc$ $x + 4$
$\bigcirc$ $x - 6$
$\bigcirc$ $x + 6$

Explanation:

Step1: Recall factoring quadratic

A quadratic \(x^2 + bx + c\) factors as \((x + m)(x + n)\) where \(m + n = b\) and \(m \times n = c\). Here, we know one factor is \(x - 9\), so let the other be \(x + a\). Then \((x - 9)(x + a)=x^2+(a - 9)x-9a\).

Step2: Compare with given quadratic

Given quadratic is \(x^2 - 5x - 36\). So, we have two equations:

1. Coefficient of \(x\): \(a - 9=-5\)
2. Constant term: \(-9a=-36\)

From the second equation: \(-9a=-36\) \(\Rightarrow\) \(a = \frac{-36}{-9}=4\).
Check with first equation: \(a - 9 = 4 - 9=-5\), which matches. So the other factor is \(x + 4\).

Answer:

\(x + 4\) (corresponding to the option "x + 4")