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mrs_skeptik [129]
3 years ago
5

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

Mathematics
2 answers:
Verdich [7]3 years ago
4 0

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.

Learn more:

1. Learn more about inverse of the function <u>brainly.com/question/1632445. </u>

2. Learn more about equation of circle <u>brainly.com/question/1506955. </u>

3. Learn more about range and domain of the function <u>brainly.com/question/3412497 </u>

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.

Paul [167]3 years ago
3 0

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.

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