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

obrozenecy6

obrozenecy6

Answered

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(x)=x35x225x+125.

Answer & Explanation

temnimam2

temnimam2

Expert

2022-01-01Added 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(x)=x35x225x+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(x)=(x5)(x225)
f(x)=(x5)(x5)(x+5)
f(x)=(x5)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 is 2
Karen Robbins

Karen Robbins

Expert

2022-01-02Added 49 answers

by grouping method:
f(x)=x35x225x+125.
=x2(x5)25(x5)
f(x)=(x5)(x225)=(x5)(x5)(x+5)
f(x)=(x5)2(x+5)
karton

karton

Expert

2022-01-04Added 439 answers

Given that the function
f(x)=x35x225x+125
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
f(x)=x2(x5)25(x5)
=(x225)(x5)
=(x+5)(x5)2
We find that the roots are -5,5,5
Or -5 with a multiplicity of 1 and 5 with a multiplicity of 2 are the roots of the equation

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