To solve for 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 find the values of [tex]\( x \)[/tex] that satisfy this equation. The function is zero when any of its factors are zero.
First, let's identify the factors and set them equal to zero:
[tex]\[ f(x) = (x-6)^2(x+2)^2 = 0 \][/tex]
The function will be zero if either of these factors is zero:
1. [tex]\((x-6)^2 = 0\)[/tex]
2. [tex]\((x+2)^2 = 0\)[/tex]
Solving these equations individually:
1. [tex]\((x-6)^2 = 0\)[/tex]
Solving [tex]\((x-6)^2 = 0\)[/tex], we take the square root of both sides:
[tex]\[ x - 6 = 0 \][/tex]
Therefore,
[tex]\[ x = 6 \][/tex]
The factor [tex]\((x-6)^2\)[/tex] indicates that the root [tex]\( x = 6 \)[/tex] has multiplicity 2 because the exponent of the factor is 2.
2. [tex]\((x+2)^2 = 0\)[/tex]
Solving [tex]\((x+2)^2 = 0\)[/tex], we take the square root of both sides:
[tex]\[ x + 2 = 0 \][/tex]
Therefore,
[tex]\[ x = -2 \][/tex]
Similarly, the factor [tex]\((x+2)^2\)[/tex] indicates that the root [tex]\( x = -2 \)[/tex] has multiplicity 2 because the exponent of the factor is 2.
Thus, the roots of the function [tex]\( f(x) = (x-6)^2(x+2)^2 \)[/tex] are:
- The root [tex]\( x = 6 \)[/tex] with multiplicity 2.
- The root [tex]\( x = -2 \)[/tex] with multiplicity 2.
Hence, the correct answers are:
- [tex]\( 6 \)[/tex] with multiplicity 2
- [tex]\(-2 \)[/tex] with multiplicity 2
