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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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3 years ago
the length of a rectangle exceeds its width by 17 inches, and the area is 18 square inches. what are the length and width of the
dimaraw [331]

The length and width of the rectangle are 18 in and 1 in, respectively.

<h3>Quadrilaterals</h3>

There are different types of quadrilaterals, for example, square, rectangle, rhombus, trapezoid, and parallelogram.  Each type is defined accordingly to its length of sides and angles. For example, in a rectangle,  the opposite sides are equal and parallel and their interior angles are equal to 90°.

The area of a rectangle can be found for the formula : b*h, where b = base and h =height.

For this question, the length exceeds its width by 17 inches - L=W+17. Here, the length is the base and the width is the height. Thus,  from the value of area given, you can find the values of the length and width of the rectangle.

A=b*h

18=(W+17)*W

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Solving this quadratic equation, you have:

w_{1,\:2}=\frac{-17\pm \sqrt{17^2-4\cdot \:1\cdot \left(-18\right)}}{2\cdot \:1}

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First using distributive property to expand the first term:

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