Answer:
\(4x^4 + 2x^3 -2x + 4\) is a prime polynomial.
Step-by-step explanation:
Given some polynomials we have to choose the polynomial which is prime.
A prime polynomial is a polynomial with integer coefficients that cannot be factored lower degree polynomials or simply we can say prime polynomial is a polynomial that can't be factored.
Option A: \(x^3+3x^2-2x-6\)
\(x^3+3x^2-2x-6=x^2(x+3)-2(x+3)=(x^2-2)(x+3)\)
∴ can be factored as \((x+3)(x^2-2)\)
Option B: \(x^3+3x^2-2x-6\)
\(x^3+3x^2-2x-6=x^2(x+3)-2(x+3)=(x^2-2)(x+3)\)
∴ can be factored as \((x^2-2)(x+3)\)
Option C: \(4x^4+4x^3-2x-2\)
\(4x^4+4x^3-2x-2=4x^3(x+1)-2(x+1)=2(x+1)(2x^3-1)\)
∴ can be factored as \(2(x+1)(2x^3-1)\)
Option D: \(4x^4 + 2x^3 -2x + 4\)
The above polynomial can't be factored.
Hence, \(4x^4 + 2x^3 -2x + 4\) is a prime polynomial.
Option D is correct.
