Answer
$f(x)= 16x^4-80x^3+32x^2+128x$
Work Step by Step
Step 1. Use the given zeros of the polynomial and write a general form $f(x)=a(x+1)(x)(x-2)(x-4)$ where $a$ is unknown.
Step 2. Use the point on the function to get $f(\frac{1}{2})=a(\frac{1}{2}+1)(\frac{1}{2})(\frac{1}{2}-2)(\frac{1}{2}-4)=63$ which gives $a=16$
Step 3. The function is $f(x)=16(x+1)(x)(x-2)(x-4)=16x^4-80x^3+32x^2+128x$
