Find the zeros of the polynomial function, and state the multiplicity of each.
\[
f(x)=(x+5)^{4}(x-4)
\]

The zeros are $-5,4$.
(Use a comma to separate answers.)
The smaller zero has multiplicity

Step 1 ：Set the function \(f(x) = (x+5)^4 * (x-4)\) equal to zero: \(0 = (x+5)^4 * (x-4)\)

Step 2 ：This equation will be true if either \((x+5)^4 = 0\) or \((x-4) = 0\)

Step 3 ：Solving for x in each case:

Step 4 ：For \((x+5)^4 = 0\), taking the fourth root of both sides, we get \(x+5 = 0\), so \(x = -5\)

Step 5 ：For \((x-4) = 0\), we get \(x = 4\)

Step 6 ：Therefore, the zeros of the function are \(x = -5\) and \(x = 4\)

Step 7 ：The multiplicity of a zero is the number of times it appears as a root, which is given by the exponent of the factor in the polynomial

Step 8 ：In this case, the factor \((x+5)\) has an exponent of 4, so the zero \(x = -5\) has a multiplicity of 4

Step 9 ：The factor \((x-4)\) has an exponent of 1 (since no exponent is shown, it is understood to be 1), so the zero \(x = 4\) has a multiplicity of 1

Step 10 ：So, the zero -5 has a multiplicity of 4 and the zero 4 has a multiplicity of 1

Step 11 ：Therefore, the final answer is \(\boxed{x = -5, 4}\) with multiplicities \(\boxed{4, 1}\) respectively