Answer
See graph and explanations.
Work Step by Step
Step 1. Use synthetic division to find three zeros as $x=3, 3, -2$ as shown in the figure.
Step 2. Factor the function as $f(x)=(x-3)^2(x+2)(-4x^2+1)=-(2x+1)(2x-1)(x-3)^2(x+2)$
Step 3. We can identify the zeros as
$x=-2$ with multiplicity 1 (curve crosses the x-axis),
$x=-\frac{1}{2}$ with multiplicity 1 (curve crosses the x-axis),
$x=\frac{1}{2}$ with multiplicity 1 (curve crosses the x-axis),
$x=3$ with multiplicity 2 (curve touches the x-axis and turns around),
Step 4. The y-intercept $f(x)=18$
Step 5. The lead term is $-4x^5$, the end behavior is rise to the left and fall to the right.
Step 6. Use test points as necessary to graph the function as shown in the figure.
