Respuesta :

The prime polynomial function is [tex]\boxed{10{x^2} - 4x + 3x + 6}[/tex]. Option C is correct.

Further explanation:

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

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

The polynomial function has n roots or zeroes.

Given:

The options are as follows,

(a). [tex]{x^3} + 3{x^2} + 2x + 6[/tex]

(b). [tex]{x^3} + 3{x^2} - 2x - 6[/tex]

(c). [tex]10{x^2} - 4x + 3x + 6[/tex]

(d). [tex]10{x^2} - 10x + 6x - 6[/tex]

Explanation:

From the graph of option (a) it has been observed that the polynomial [tex]{x^3} + 3{x^2} + 2x + 6[/tex] has one root at [tex]x =  - 3[/tex]. Therefore, it is not a prime polynomial function.

From the graph of option (b) it has been observed that the polynomial [tex]{x^3} + 3{x^2} - 2x - 6[/tex] has two roots at [tex]x =  - 3[/tex] and [tex]x = -1.414[/tex]. Therefore, it is not a prime polynomial function.

From the graph of option (c) it has been observed that the polynomial [tex]10{x^2} - 4x + 3x + 6[/tex] has no root. Therefore, it is a prime polynomial function.

From the graph of option (d) it has been observed that the polynomial [tex]10{x^2} - 10x + 6x - 6[/tex] has two roots. Therefore, it is not a prime polynomial function.

The prime polynomial function is [tex]\boxed{10{x^2} - 4x + 3x + 6}[/tex]. Option C is correct.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Polynomial

Keywords: roots, prime polynomial, linear equation, quadratic equation, zeros, function, polynomial, solution, cubic function, degree of the function.

okpalawalter8

We have that  the Prime Polynomial in the midst of the options is

  • Option 3

10x2 – 4x + 3x + 6=2x(5x-2)+3(x+2)

Hence it cant be factored

  • it is Prime

From the question we are told

Which polynomial is prime?

  • x3 + 3x2 + 2x + 6
  • x3 + 3x2 – 2x – 6
  • 10x2 – 4x + 3x + 6
  • 10x2 – 10x + 6x – 6

Prime Polynomial

Generally Prime Polynomial is said to occur when a polynomial can't be factored with integer,rational or real coefficient

Where

Option A

x3 + 3x2 + 2x + 6 Not prime

Option B

x3 + 3x2 – 2x – 6 not Prime

Option C

10x2 – 10x + 6x – 6 Not Prime

Therefore

Option 3

10x2 – 4x + 3x + 6=2x(5x-2)+3(x+2)

Hence it cant be factored

it is Prime

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