College Algebra, Chapter 4, 4.3, Section 4.3, Problem 68

Determine a polynomial of degree $4$ whose graph is shown







Based from the graph, if the graph crosses the $x$-axis at $x = -2$, then it has a single root $(0)$ at $x = -2$, so $x + 2$ is a factor. Similarly, the graph crosses the $x$-axis at $x = -1$, then it has a single root $(0)$ at $x = -1$, so $x + 1$ is a factor, and if the graph bounces off the $x$-axis at $x = 1$, then there must be a double root $(0)$ at $x = 1$, so $(x -1)^2$ is a factor. We let


$
\begin{equation}
\begin{aligned}

P(x) =& (x + 2)(x + 1)(x - 1)^2
\\
\\
P(x) =& (x^2 + x + 2x + 2)(x^2 - 2x + 1)
\\
\\
P(x) =& (x^2 + 3x + 2)(x^2 - 2x + 1)
\\
\\
P(x) =& x^4 - 2x^3 + x^2 + 3x^3 - 6x^2 + 3x + 2x^2 - 4x + 2
\\
\\
P(x) =& x^4 + x^3 - 3x^2 - x + 2

\end{aligned}
\end{equation}
$