Answered

Join IDNLearn.com today and start getting the answers you've been searching for. Get the information you need quickly and accurately with our reliable and thorough Q&A platform.

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 :

GinnyAnsweridn
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.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.