Question

According to the Fundamental Theorem of Algebra, which polynomial function has exactly 6 roots?

a. f(x)= 5x^4+10x^2

b. f(x)= 5x^5+3x^4+12x^3+7x^2-2x+15

c. f(x)= 6x^5+x^3-4x^2+x-5

d. f(x)= 7x^6+3x^3+12

Answer 1

Answer: d.$f(x)=7x^6+3x^3+12$

Explanation:-

• The fundamental theorem of algebra tells that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
• Corollary to the fundamental theorem tells that every polynomial of m>0 degree has exactly m zeroes.

Thus only option d.$f(x)=7x^6+3x^3+12$ with degree 6 has exactly 6 roots.

Answer 2

Option (D) is correct as the polynomial $f\left(x\right)=7{x^6}+3{x^3}+12$ has exactly 6 roots according to the Fundamental Theorem of Algebra.

Further explanation:

The Fundamental Theorem of Algebra states that the polynomial has n roots if the degree of the polynomial is n.

$f\left(x\right)=a{x^n}+b{x^{n-1}}+\ldots+cx+d$

The polynomial function has n roots or zeroes.

Given:

The polynomial functions are as follows.

(a). $f\left(x\right)=5{x^4}+10{x^2}$

(b). $f\left(x\right)=5{x^5}+3{x^4}+12{x^3}+7{x^2}-2x+15$

(c). $f\left(x\right)=6{x^5}+{x^3}-4{x^2}+x-5$

(d). $f\left(x\right)=7{x^6}+3{x^3}+12$

Explanation:

The polynomial $f\left(x\right)=5{x^4}+10{x^2}$ has maximum 4 roots as the degree of the polynomial is 4.

Hence, option (a) is not correct.

The polynomial $f\left(x\right)=5{x^5}+3{x^4}+12{x^3}+7{x^2}-2x+15$ has maximum 5 roots as the degree of the polynomial is 5.

Hence, option (b) is not correct.

The polynomial $f\left( x \right) = 6{x^5} + {x^3} - 4{x^2}$ + x – 5 has maximum 5 roots as the degree of the polynomial is 5.

Hence, option (c) is not correct.

The polynomial $f\left(x\right)=7{x^6}+3{x^3}+12$ has maximum 6 roots as the degree of the polynomial is 6.

Hence, Option (d) is correct as the polynomial $f\left(x\right)=7{x^6}+3{x^3}+12$ has exactly 6 roots according to the Fundamental Theorem of Algebra.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Polynomials

Keywords: quadratic equation, equation factorization. Factorized form, polynomial, quadratic formula, zeroes, Fundamental Theorem of algebra, polynomial, six roots.