To find the zeros of the polynomial function [tex]\( f(x) = -4(x-4)^2(x+5)^3 x^4 \)[/tex] and determine their multiplicities, follow these steps:
1. Identify and factorize the polynomial: The polynomial is already factored in the form [tex]\( f(x) = -4(x-4)^2(x+5)^3 x^4 \)[/tex].
2. Find the zeros:
- Factor: [tex]\((x-4)^2\)[/tex]
- Zero: [tex]\(x = 4\)[/tex]
- Multiplicity: 2 (since the factor [tex]\((x-4)\)[/tex] is squared)
- Factor: [tex]\((x+5)^3\)[/tex]
- Zero: [tex]\(x = -5\)[/tex]
- Multiplicity: 3 (since the factor [tex]\((x+5)\)[/tex] is cubed)
- Factor: [tex]\( x^4\)[/tex]
- Zero: [tex]\(x = 0\)[/tex]
- Multiplicity: 4 (since the factor [tex]\(x\)[/tex] is raised to the fourth power)
Given these steps, we identify three zeros and their respective multiplicities.
The smallest zero is [tex]\( -5 \)[/tex] with multiplicity 3.
The middle zero is [tex]\( 0 \)[/tex] with multiplicity 4.
The largest zero is [tex]\( 4 \)[/tex] with multiplicity 2.
So, we can fill in the blanks as follows:
The smallest zero is [tex]\(-5\)[/tex] with multiplicity [tex]\(3\)[/tex].
The middle zero is [tex]\(0\)[/tex] with multiplicity [tex]\(4\)[/tex].
The largest zero is [tex]\(4\)[/tex] with multiplicity [tex]\(2\)[/tex].
