College Algebra, Chapter 4, Chapter Review, Section Review, Problem 42

If $P(x) = x^4 - 5x^2 + 4$, 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^4 - 5x2 + 4
&&
\\
\\
=& (x^2 - 4)(x^2 - 1)
&& \text{Factor}
\\
\\
=& (x + 2)(x - 2)(x + 1)(x - 1)
&& \text{Difference of squares}

\end{aligned}
\end{equation}
$



It shows that the function has zeros of $-2, 2, -1$ 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 $-2, 2, -1$ and $1$. To determine the $y$ intercept, we set $x = 0$ so, $P(0) = ( 0 + 2)(0 - 2)(0 + 1)(0 - 1) = (2)(-2)(1)(-1) = 4$

Since the function has an even 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