Answer
$f(x)=-\frac{1}{4}(x+1)^2(x-1)(x-4)$
Work Step by Step
Step 1. Based on the given graph, we can identify the real zeros as $x=-1$ (multiplicity 2), $x=1$ (multiplicity 1), and $x=4$ (multiplicity 1).
Step 2. We can write a general form of the polynomial as $f(x)=a(x+1)^2(x-1)(x-4)$ where $a$ is unknown.
Step 3. Use the given point $(3,8)$ on the curve, we have $f(3)=a(3+1)^2(3-1)(3-4)=8$, thus $a=-\frac{1}{4}$
Step 4. Thus the function is $f(x)=-\frac{1}{4}(x+1)^2(x-1)(x-4)$
