obrozenecy6

2021-12-31

Using the Fundamental Theorem of Algebra, complete the following exercise. Show your work.

Determine how many, what type, and find the roots for$f\left(x\right)={x}^{3}-5{x}^{2}-25x+125$ .

Determine how many, what type, and find the roots for

temnimam2

Beginner2022-01-01Added 36 answers

Step 1

Fundamental Theorem of Algebra, states if f(x) is a polynomial of degree n, where$n>0$ , then f has at least one zero in the complex number system.

The polynomial given to us is$f\left(x\right)={x}^{3}-5{x}^{2}-25x+125$ .

Step 2

the highest power of this polynomial is 3 so, there will be 3 roots to this polynomial.

now, by factorizing the polynomial we get

$f\left(x\right)=(x-5)({x}^{2}-25)$

$f\left(x\right)=(x-5)(x-5)(x+5)$

$f\left(x\right)={(x-5)}^{2}(x+5)$

therefore, we can see that the roots of the polynomial are real and the roots of the polynomial are$x=5,-5$

and the multiplicity of the roots$x=-5\text{}is\text{}2$

Fundamental Theorem of Algebra, states if f(x) is a polynomial of degree n, where

The polynomial given to us is

Step 2

the highest power of this polynomial is 3 so, there will be 3 roots to this polynomial.

now, by factorizing the polynomial we get

therefore, we can see that the roots of the polynomial are real and the roots of the polynomial are

and the multiplicity of the roots

Karen Robbins

Beginner2022-01-02Added 49 answers

by grouping method:

$f\left(x\right)={x}^{3}-5{x}^{2}-25x+125$ .

$={x}^{2}(x-5)-25(x-5)$

$f\left(x\right)=(x-5)({x}^{2}-25)=(x-5)(x-5)(x+5)$

$f\left(x\right)={(x-5)}^{2}(x+5)$

karton

Expert2022-01-04Added 613 answers

Given that the function

We have to find the roots of f(x)

Let us try to factorize the function to find the roots

We can group two by two and find out

We find that the roots are -5,5,5

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