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Find the root(s) of [tex]\( f(x) = (x-6)^2(x+2)^2 \)[/tex].

A. -6 with multiplicity 1
B. -6 with multiplicity 2
C. 6 with multiplicity 1
D. 6 with multiplicity 2
E. -2 with multiplicity 1
F. -2 with multiplicity 2
G. 2 with multiplicity 1
H. 2 with multiplicity 2


Sagot :

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