Question

Which polynomial is prime?

Answer 1

we know that

If the only factors a polynomial are $1$ and itself, then that polynomial is prime.

so

If the polynomial has at least one root (one x-intercept) then the polynomial is not prime

Using a graph tool

case 1)

$x^{3}+ 3x^{2} + 2x+ 6$

see the attached figure N 1

The polynomial has one root-------> the polynomial is not prime

case 2)

$x^{3} + 3x^{2} - 2x - 6$

see the attached figure N 2

The polynomial has three roots-------> the polynomial is not prime

case 3)

$10x^{2}-4x+3x + 6$

$10x^{2} -x + 6$

see the attached figure N 3

The polynomial has zero roots-------> the polynomial is  prime

case 4)

$10x^{2}-10x+6x - 6$

$10x^{2} -4x - 6$

see the attached figure N 4

The polynomial has two roots-------> the polynomial is not prime

therefore

the answer is

$10x^{2}-4x+3x + 6$ ------> is prime

Answer 2

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

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 options are as follows,

(a). ${x^3} + 3{x^2} + 2x + 6$

(b). ${x^3} + 3{x^2} - 2x - 6$

(c). $10{x^2} - 4x + 3x + 6$

(d). $10{x^2} - 10x + 6x - 6$

Explanation:

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

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

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

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

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

Learn more:

1. Learn more about inverse of the function brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function brainly.com/question/3412497.

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.