College Algebra, Chapter 4, Chapter Review, Section Review, Problem 52

Find all rational, irrational and complex zeros (and state their multiplicities) of the polynomial function $P(x) = 2x^3 + 5x^2 - 6x - 9$. Use Descartes' Rule of signs, the Upper and Lower Bounds Theorem, the Quadratic Formula or other factoring techniques.

The possible rational zeros of the polynomial $P$ is the factor of $9$ divided by the factor of the leading coefficient $2$ which are $\displaystyle \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}$ and $\displaystyle \pm \frac{9}{2}$. By using Synthetic Division and trial and error







Again, by applying Synthetic Division







Thus,


$
\begin{equation}
\begin{aligned}

P(x) =& 2x^3 + 5x^2 - 6x - 9
\\
\\
=& (x + 3)(2x^2 - x - 3)
\\
\\
=& (x + 3)(x + 1)(2x - 3)

\end{aligned}
\end{equation}
$


Therefore, the zeros of $P$ are $-3, -1$ and $\displaystyle \frac{3}{2}$. Each zeros have multiplicity of $1$.