Answer
$f(x)= -\frac{16}{49}x^4+\frac{32}{49}x^3+\frac{496}{49}x^2-\frac{512}{49}x-\frac{960}{49}$
Work Step by Step
Step 1. Use the given zeros of the polynomial and write a general form $f(x)=a(x+5)(x+1)(x-2)(x-6)$ where $a$ is unknown.
Step 2. Use the point on the function to get $f(\frac{5}{2})=a(\frac{5}{2}+5)(\frac{5}{2}+1)(\frac{5}{2}-2)(\frac{5}{2}-6)=15$ which gives $a=-\frac{16}{49}$
Step 3. The function is $f(x)=-\frac{16}{49}(x+5)(x+1)(x-2)(x-6)=-\frac{16}{49}x^4+\frac{32}{49}x^3+\frac{496}{49}x^2-\frac{512}{49}x-\frac{960}{49}$
