College Algebra, Chapter 4, Chapter Review, Section Review, Problem 54

Find all rational, irrational and complex zeros (and state their multiplicities) of the polynomial function $P(x) = x^4 + 7x^3 + 9x^2 - 17x - 20$. 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 $P$ are the factors of $20$ which are $\displaystyle \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$. Then, by using Synthetic Division and trial and error







Again, by applying Synthetic Division







Thus,


$
\begin{equation}
\begin{aligned}

P(x) =& x^4 + 7x^3 + 9x^2 - 17x - 20
\\
\\
=& (x + 4)(x^3 + 3x^2 - 3x - 5)
\\
\\
=& (x + 4)(x + 1)(x^2 + 2x - 5)


\end{aligned}
\end{equation}
$


To get the remaining zeros, we use quadratic formula.


$
\begin{equation}
\begin{aligned}

x =& \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\\
\\
=& \frac{-(2) \pm \sqrt{2^2 - 4(1) (-5)}}{2(1)}
\\
\\
=& -1 \pm \sqrt{6}

\end{aligned}
\end{equation}
$


Therefore, the zeros of $P$ are $-4, -1, -1 + \sqrt{6}$ and $-1 - \sqrt{6}$. Each zeros have multiplicity of $1$.