Get detailed and accurate answers to your questions on IDNLearn.com. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.

Which of the following describes the roots of the polynomial function [tex]$f(x) = (x - 3)^4 (x + 6)^2$[/tex]?

A. -3 with multiplicity 2 and 6 with multiplicity 4
B. -3 with multiplicity 4 and 6 with multiplicity 2
C. 3 with multiplicity 2 and -6 with multiplicity 4
D. 3 with multiplicity 4 and -6 with multiplicity 2


Sagot :

To determine the roots and their multiplicities for the polynomial function [tex]\(f(x) = (x-3)^4(x+6)^2\)[/tex], we need to examine the factors of the polynomial.

1. Identify the roots from the factors:
- The factor [tex]\((x-3)^4\)[/tex] implies a root at [tex]\(x = 3\)[/tex].
- The factor [tex]\((x+6)^2\)[/tex] implies a root at [tex]\(x = -6\)[/tex].

2. Determine the multiplicities of the roots:
- The exponent on the factor [tex]\((x-3)^4\)[/tex] is 4, which indicates that the root [tex]\(x = 3\)[/tex] has a multiplicity of 4.
- The exponent on the factor [tex]\((x+6)^2\)[/tex] is 2, which indicates that the root [tex]\(x = -6\)[/tex] has a multiplicity of 2.

Now, match these observations to the given choices:

1. [tex]\(-3\)[/tex] with multiplicity 2 and [tex]\(6\)[/tex] with multiplicity 4
- This does not match our roots and their multiplicities.

2. [tex]\(-3\)[/tex] with multiplicity 4 and [tex]\(6\)[/tex] with multiplicity 2
- This also does not match our roots and their multiplicities.

3. [tex]\(3\)[/tex] with multiplicity 2 and [tex]\(-6\)[/tex] with multiplicity 4
- This does not match our roots and their multiplicities either.

4. [tex]\(3\)[/tex] with multiplicity 4 and [tex]\(-6\)[/tex] with multiplicity 2
- This matches our roots and their multiplicities exactly.

Thus, the correct description of the roots of the polynomial function [tex]\(f(x) = (x-3)^4(x+6)^2\)[/tex] is:
[tex]\[ 3 \text{ with multiplicity 4 and } -6 \text{ with multiplicity 2} \][/tex]

Therefore, the correct choice is:
[tex]\[ \boxed{4} \][/tex]