To determine the roots and their multiplicities for the function [tex]\( f(x) = (x-6)^2(x+2)^2 \)[/tex], follow these steps:
1. Identify the Factors and Their Zeroes:
- The function [tex]\( f(x) = (x-6)^2(x+2)^2 \)[/tex] is composed of two distinct factors.
- These factors are [tex]\( (x-6)^2 \)[/tex] and [tex]\( (x+2)^2 \)[/tex].
2. Set Each Factor to Zero and Solve for [tex]\( x \)[/tex]:
- For the factor [tex]\( (x-6)^2 \)[/tex]:
[tex]\[
(x-6)^2 = 0
\][/tex]
Taking the square root of both sides:
[tex]\[
x-6 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 6
\][/tex]
- For the factor [tex]\( (x+2)^2 \)[/tex]:
[tex]\[
(x+2)^2 = 0
\][/tex]
Taking the square root of both sides:
[tex]\[
x+2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -2
\][/tex]
3. Determine the Multiplicities:
- The multiplicity of a root is determined by the exponent of the factor corresponding to that root.
- For the root [tex]\( x = 6 \)[/tex]:
- The factor [tex]\( (x-6)^2 \)[/tex] indicates the root [tex]\( x = 6 \)[/tex] with multiplicity 2.
- For the root [tex]\( x = -2 \)[/tex]:
- The factor [tex]\( (x+2)^2 \)[/tex] indicates the root [tex]\( x = -2 \)[/tex] with multiplicity 2.
4. State the Final Results:
- The roots and their multiplicities are:
- [tex]\( x = 6 \)[/tex] with multiplicity 2.
- [tex]\( x = -2 \)[/tex] with multiplicity 2.
So, the correct answers are:
- 6 with multiplicity 2
- -2 with multiplicity 2
