#### Answer

$f(x)=x^3-\frac{7}{2}x^2-2x$

#### Work Step by Step

RECALL:
If $a$ is a zero of a polynomial, then $x−a$ is a factor of the polynomial.
Thus, having $x=0, 4, -\frac{1}{2}$ as zeros of the polynomial function means that $x,x-4$, and $x+\frac{1}{2}$ are factors of the polynomial.
Therefore the function is
$f(x)=x(x-4)(x+\frac{1}{2})
\\f(x)=x^3-\frac{7}{2}x^2-2x.$