Answer:
[tex]4x^4 + 2x^3 -2x + 4[/tex] 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: [tex]x^3+3x^2-2x-6[/tex]
[tex]x^3+3x^2-2x-6=x^2(x+3)-2(x+3)=(x^2-2)(x+3)[/tex]
∴ can be factored as [tex](x+3)(x^2-2)[/tex]
Option B: [tex]x^3+3x^2-2x-6[/tex]
[tex]x^3+3x^2-2x-6=x^2(x+3)-2(x+3)=(x^2-2)(x+3)[/tex]
∴ can be factored as [tex](x^2-2)(x+3)[/tex]
Option C: [tex]4x^4+4x^3-2x-2[/tex]
[tex]4x^4+4x^3-2x-2=4x^3(x+1)-2(x+1)=2(x+1)(2x^3-1)[/tex]
∴ can be factored as [tex]2(x+1)(2x^3-1)[/tex]
Option D: [tex]4x^4 + 2x^3 -2x + 4[/tex]
The above polynomial can't be factored.
Hence, [tex]4x^4 + 2x^3 -2x + 4[/tex] is a prime polynomial.
Option D is correct.
