Find the best solutions to your problems with the help of IDNLearn.com's experts. Discover trustworthy solutions to your questions quickly and accurately with help from our dedicated community of experts.

Determine the multiplicity of the roots of the function [tex]k(x) = x(x+2)^3(x+4)^2(x-5)^4[/tex].

- 0 has multiplicity:
- -2 has multiplicity:
- -4 has multiplicity:
- 5 has multiplicity:


Sagot :

To determine the multiplicity of the roots of the function [tex]\(k(x) = x(x + 2)^3(x + 4)^2(x - 5)^4\)[/tex], we will examine each factor of the function separately.

1. Root at [tex]\(x = 0\)[/tex]:
- The factor that gives the root [tex]\(x = 0\)[/tex] is [tex]\(x\)[/tex].
- Since this factor is of the form [tex]\(x\)[/tex], the exponent is 1.
- Therefore, the root [tex]\(x = 0\)[/tex] has multiplicity 1.

2. Root at [tex]\(x = -2\)[/tex]:
- The factor that gives the root [tex]\(x = -2\)[/tex] is [tex]\((x + 2)^3\)[/tex].
- Since this factor is of the form [tex]\((x + 2)^3\)[/tex], the exponent is 3.
- Therefore, the root [tex]\(x = -2\)[/tex] has multiplicity 3.

3. Root at [tex]\(x = -4\)[/tex]:
- The factor that gives the root [tex]\(x = -4\)[/tex] is [tex]\((x + 4)^2\)[/tex].
- Since this factor is of the form [tex]\((x + 4)^2\)[/tex], the exponent is 2.
- Therefore, the root [tex]\(x = -4\)[/tex] has multiplicity 2.

4. Root at [tex]\(x = 5\)[/tex]:
- The factor that gives the root [tex]\(x = 5\)[/tex] is [tex]\((x - 5)^4\)[/tex].
- Since this factor is of the form [tex]\((x - 5)^4\)[/tex], the exponent is 4.
- Therefore, the root [tex]\(x = 5\)[/tex] has multiplicity 4.

In summary:
- The root [tex]\(0\)[/tex] has multiplicity 1.
- The root [tex]\(-2\)[/tex] has multiplicity 3.
- The root [tex]\(-4\)[/tex] has multiplicity 2.
- The root [tex]\(5\)[/tex] has multiplicity 4.