College Algebra, Chapter 4, Chapter Review, Section Review, Problem 14

Graph the polynomial $P(x) = -3 (x + 2)^5 + 96$ by transforming an appropriate graph of the form $y = x^5$ and show all the $x$ and $y$ intercepts clearly.

The graph of $P(x) = -3 (x + 2)^5 + 96$ is obtained from the graph of $y = x^5$ that is shifted $2$ units to the left,
reflected about the $x$ axis and stretched vertically by a factor of $3$. Then, the result is shifted $96$ units upward. To determine the $x$ intercept, we set $y = 0$ so


$
\begin{equation}
\begin{aligned}

0 =& -3 (x + 2)^5 + 96
\\
\\
3 (x + 2)^5 =& 96
\\
\\
(x + 2)^5 =& 32
\\
\\
x + 2 =& \sqrt[5]{32}
\\
\\
x + 2 =& 2
\\
\\
x =& 0


\end{aligned}
\end{equation}
$


Next, to determine the $y$ intercept, we set $x = 0$.


$
\begin{equation}
\begin{aligned}

P(0) =& -3 (0 + 2)^5 + 96
\\
\\
=& -3(32) + 96
\\
\\
=& -96 + 96
\\
\\
=& 0

\end{aligned}
\end{equation}
$


The $y$ intercept is .