Find a polynomial function of degree 4 with the zeros -3 (multiplicity 2 ) and 3 (multiplicity 2), whose graph passes through the point $(-4,196)$.
\[
f(x)=
\]
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
Answer
So, the polynomial function of degree 4 with the zeros -3 (multiplicity 2 ) and 3 (multiplicity 2), whose graph passes through the point \((-4,196)\) is \(\boxed{f(x)= 4(x+3)^2(x-3)^2}\).
Step 1 ：The polynomial function of degree 4 with the given zeros can be written in the form of \(f(x) = a(x+3)^2(x-3)^2\).
Step 2 ：To find the value of \(a\), we can substitute the point \((-4,196)\) into the function and solve for \(a\).
Step 3 ：After solving, we find that \(a = 4\).
Step 4 ：So, the polynomial function of degree 4 with the zeros -3 (multiplicity 2 ) and 3 (multiplicity 2), whose graph passes through the point \((-4,196)\) is \(\boxed{f(x)= 4(x+3)^2(x-3)^2}\).
