College Algebra, Chapter 4, 4.3, Section 4.3, Problem 62

Find a polynomial of degree $5$ that has zeros $-2, -1, 0, 1, 2$.

By the factor theorem $x - (-2), x - (-1), x - 0, x - 1$ and $x - 2$ must all be factors of the desired polynomial, so let


$
\begin{equation}
\begin{aligned}

P(x) =& (x + 2)(x + 1) (x - 0)(x - 1)(x - 2)
\\
\\
P(x) =& (x^2 - 4) (x^2 - 1)(x)
\\
\\
P(x) =& (x^4 - x^2 - 4x^2 + 4)(x)
\\
\\
P(x) =& (x^4 - 5x^2 + 4)(x)
\\
\\
P(x) =& x^5 - 5x^3 + 4x

\end{aligned}
\end{equation}
$


Since $P(x)$ is of degree $5$, it is a solution of the problem.