College Algebra, Chapter 4, Chapter Review, Section Review, Problem 40

If $P(x) = x^3 - 3x^2 - 4x$, then

a.) Find all zeros of $P$, and state their multiplicities.

b.) Sketch the graph of $P$.



a.) To find the zeros of $P$, we factor $P$ to obtain


$
\begin{equation}
\begin{aligned}

P(x) =& x^3 - 3x^2 - 4x
&& \text{Given}
\\
\\
=& x (x^2 - 3x - 4)
&& \text{Factor out } x
\\
\\
=& x (x - 4)(x + 1)
&& \text{Factor the quadratic function}

\end{aligned}
\end{equation}
$



It shows that the function has zeros of $0, 4$ and $-1$. And all the zeros have multiplicity of $1$.

b.) To sketch the graph of $P$, we must know first the intercepts of the function. The values of the $x$ intercepts are the zeros of the function, that is $0, 4$ and $-1$. To determine the $y$ intercept, we set $x = 0$ so, $P(0) = 0(0 - 4)(0 + 1) = 0$

The $y$ intercept is .

Since the function has an odd degree and a positive leading coefficient, then its end behavior is $y \to \infty$ as $x \to \infty$ and $y \to - \infty$ as $x \to - \infty$. Then, the graph is