Answer
$P(x)=(2x-1)(x-3)(x+3)$
Zeros: $-3, \frac{1}{2}, \text{ and } 3$
Refer to the graph below.
Work Step by Step
Factor the polynomial completely to obtain:
\begin{align*}
P(x)&=(2x^3-x^2)+(-18x+9)\\
&=x^2(2x-1)+(-9)(2x-1)\\
&=(2x-1)(x^2-9)\\
&=(2x-1)(x-3)(x+3)\\
\end{align*}
To find the zeros, use the Zero-Product Property by equating each factor to $0$, then solve each equation to obtain:
\begin{align*}
2x-1&=0 &\text{or}& &x-3=0& &\text{or}& =&x+3=0\\
x&=\frac{1}{2} &\text{or}& &x=3& &\text{or}& =&x=-3\\
\end{align*}
Use a graphing utility to graph $P(x)$.
Refer to the graph above.
