To find the roots of the function [tex]\( f(x) = (x-6)^2(x+2)^2 \)[/tex] along with their multiplicities, we need to analyze each factor of the polynomial.
1. Identify the factors: The function is [tex]\( f(x) = (x-6)^2(x+2)^2 \)[/tex]. The roots of the polynomial can be observed from the factors [tex]\((x-6)^2\)[/tex] and [tex]\((x+2)^2\)[/tex].
2. Determine the roots:
- From the factor [tex]\((x-6)^2\)[/tex], we observe that [tex]\(x = 6\)[/tex] is a root.
- From the factor [tex]\((x+2)^2\)[/tex], we observe that [tex]\(x = -2\)[/tex] is a root.
3. Determine the multiplicities:
- The exponent of the factor [tex]\((x-6)^2\)[/tex] indicates the multiplicity of the root [tex]\( x = 6 \)[/tex]. In this case, the exponent is 2, so [tex]\( x = 6 \)[/tex] has a multiplicity of 2.
- Similarly, the exponent of the factor [tex]\((x+2)^2\)[/tex] indicates the multiplicity of the root [tex]\( x = -2 \)[/tex]. Here, the exponent is also 2, so [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
So, summarizing the roots and their multiplicities:
- The root [tex]\( x = 6 \)[/tex] has a multiplicity of 2.
- The root [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
Therefore, the correct answers are:
- [tex]\(6\)[/tex] with multiplicity 2.
- [tex]\(-2\)[/tex] with multiplicity 2.
