To determine the roots and their multiplicities for the polynomial function [tex]\(f(x) = (x-3)^4(x+6)^2\)[/tex], we need to examine the factors of the polynomial.
1. Identify the roots from the factors:
- The factor [tex]\((x-3)^4\)[/tex] implies a root at [tex]\(x = 3\)[/tex].
- The factor [tex]\((x+6)^2\)[/tex] implies a root at [tex]\(x = -6\)[/tex].
2. Determine the multiplicities of the roots:
- The exponent on the factor [tex]\((x-3)^4\)[/tex] is 4, which indicates that the root [tex]\(x = 3\)[/tex] has a multiplicity of 4.
- The exponent on the factor [tex]\((x+6)^2\)[/tex] is 2, which indicates that the root [tex]\(x = -6\)[/tex] has a multiplicity of 2.
Now, match these observations to the given choices:
1. [tex]\(-3\)[/tex] with multiplicity 2 and [tex]\(6\)[/tex] with multiplicity 4
- This does not match our roots and their multiplicities.
2. [tex]\(-3\)[/tex] with multiplicity 4 and [tex]\(6\)[/tex] with multiplicity 2
- This also does not match our roots and their multiplicities.
3. [tex]\(3\)[/tex] with multiplicity 2 and [tex]\(-6\)[/tex] with multiplicity 4
- This does not match our roots and their multiplicities either.
4. [tex]\(3\)[/tex] with multiplicity 4 and [tex]\(-6\)[/tex] with multiplicity 2
- This matches our roots and their multiplicities exactly.
Thus, the correct description of the roots of the polynomial function [tex]\(f(x) = (x-3)^4(x+6)^2\)[/tex] is:
[tex]\[ 3 \text{ with multiplicity 4 and } -6 \text{ with multiplicity 2} \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{4} \][/tex]
