To find the roots of the function [tex]\( f(x) = (x - 6)^2 (x + 2)^2 \)[/tex], we need to set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 6)^2 (x + 2)^2 = 0. \][/tex]
By the zero-product property, for a product of factors to be zero, at least one of the factors must be zero.
1. Set each factor equal to zero and solve:
[tex]\[ (x - 6)^2 = 0 \][/tex]
[tex]\[ x - 6 = 0 \][/tex]
[tex]\[ x = 6. \][/tex]
Since the factor [tex]\((x - 6)\)[/tex] is squared, the root [tex]\( x = 6 \)[/tex] has a multiplicity of 2.
2. Now solve the other factor:
[tex]\[ (x + 2)^2 = 0 \][/tex]
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2. \][/tex]
Similarly, since the factor [tex]\((x + 2)\)[/tex] is squared, the root [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
Therefore, the roots and their multiplicities are:
- [tex]\( x = 6 \)[/tex] with multiplicity 2.
- [tex]\( x = -2 \)[/tex] with multiplicity 2.
So, the correct answers from the given choices are:
- [tex]\( 6 \)[/tex] with multiplicity 2
- [tex]\( -2 \)[/tex] with multiplicity 2
