Answer
$f(x)=(x-1)(x-6)(x+3)$
Work Step by Step
We are given the polynomial function:
$$f(x)=x^3-4x^2-15x+18.$$
$\bf{Step\text{ }1}$
First we will list the possible rational zeros. The leading coefficient is $1$ and the constant term is $18$. So the possible rational zeros are:
$$\pm 1,\pm2,\pm 3,\pm 6,\pm 9,\pm 18.$$
$\bf{Step\text{ }2}$
Test these zeros using synthetic division:
Test $x=1$:
This gives:
$$f(x)=(x-1)(x^2-3x-18).$$
$\bf{Step\text{ }3}$
Factor the polynomial using the Factor Theorem:
$$\begin{align*}
f(x)&=(x-1)((x^2-6x)+(3x-18))\\
&=(x-1)(x(x-6)+3(x-6)\\
&=(x-1)(x-6)(x+3).
\end{align*}$$
