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 $f(x)=(x+5{)}^4\ast (x-4)$ equal to zero: $0=(x+5{)}^4\ast (x-4)$

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 $\boxed{{\displaystyle x=-5,4}}$ with multiplicities $\boxed{{\displaystyle 4,1}}$ respectively