To determine which polynomial is prime, we need to check whether each polynomial can be factored into polynomials of lower degree with integer coefficients. If a polynomial cannot be factored in this manner, it is considered "prime" (or irreducible).
Let's analyze each polynomial step-by-step:
1. [tex]\( x^3 + 3x^2 - 2x - 6 \)[/tex]:
This polynomial can be checked by attempting to factor it. If it can be written as a product of polynomials of lower degrees, it is not prime. In this case, this polynomial can be factored into lower degree polynomials, revealing that it is not prime.
2. [tex]\( x^3 - 2x^2 + 3x - 6 \)[/tex]:
Similarly, for this polynomial, if it can be expressed as a product of polynomials of lower degrees, it is not prime. This polynomial can also be factored, suggesting that it is not prime.
3. [tex]\( 4x^4 + 4x^3 - 2x - 2 \)[/tex]:
We examine this polynomial for factorization. If it can be decomposed into polynomials of lower degree, then it is not prime. This polynomial can be factored, indicating that it is not prime.
4. [tex]\( 2x^4 + x^3 - x + 2 \)[/tex]:
Checking factorization for this polynomial, we find that it cannot be expressed as a product of polynomials of lower degree with integer coefficients. Therefore, this polynomial cannot be factored any further, indicating that it is prime or irreducible.
From the analysis, we observe that the only polynomial that cannot be factored is:
[tex]\[ 2x^4 + x^3 - x + 2 \][/tex]
Thus, the polynomial [tex]\( 2x^4 + x^3 - x + 2 \)[/tex] is prime.
