To determine which of the given polynomials is prime, we need to factor each polynomial and check if it can be factored into irreducible polynomials (other than 1 and itself). Let's go through each polynomial step-by-step:
1. Polynomial: [tex]\(x^4 + 3x^2 - x^2 - 3\)[/tex]
Simplify:
[tex]\[
x^4 + (3x^2 - x^2) - 3 = x^4 + 2x^2 - 3
\][/tex]
Factor:
[tex]\[
\text{(This polynomial can be factored further into irreducible factors)}
\][/tex]
2. Polynomial: [tex]\(x^4 - 3x^2 - x^2 + 3\)[/tex]
Simplify:
[tex]\[
x^4 - (3x^2 + x^2) + 3 = x^4 - 4x^2 + 3
\][/tex]
Factor:
[tex]\[
\text{(This polynomial can be factored further into irreducible factors)}
\][/tex]
3. Polynomial: [tex]\(3x^2 + x - 6x - 2\)[/tex]
Simplify:
[tex]\[
3x^2 + (x - 6x) - 2 = 3x^2 - 5x - 2
\][/tex]
Factor:
[tex]\[
\text{(This polynomial can be factored further into irreducible factors)}
\][/tex]
4. Polynomial: [tex]\(3x^2 + x - 6x + 3\)[/tex]
Simplify:
[tex]\[
3x^2 + (x - 6x) + 3 = 3x^2 - 5x + 3
\][/tex]
Factor:
[tex]\[
\text{(This polynomial is already at its simplest form and cannot be factored further)}
\][/tex]
Based on the above steps, we find that the polynomial [tex]\(3x^2 + x - 6x + 3 = 3x^2 - 5x + 3\)[/tex] cannot be factored into simpler polynomials and is therefore prime.
Conclusion:
The polynomial [tex]\(3x^2 + x - 6x + 3 \)[/tex] is the prime polynomial.
