To determine which polynomials are prime, we need to check if each polynomial cannot be factored into products of polynomials of lower degrees with coefficients in the set of rational numbers (i.e., they are irreducible over the rationals).
Let's examine each polynomial:
1. Polynomial: [tex]\(3x^3 + 3x^2 - 2x - 2\)[/tex]
- Typically, we seek to factor the polynomial by attempting to find rational roots using the Rational Root Theorem or polynomial long division.
- However, no factorization simplifies this polynomial into lower degree polynomials with rational coefficients.
- Therefore, this polynomial is prime.
2. Polynomial: [tex]\(3x^3 - 2x^2 + 3x - 4\)[/tex]
- Similar to the previous polynomial, attempts to factor this polynomial using standard methods (such as factoring by grouping or synthetic division) do not yield simpler polynomials with rational coefficients.
- Thus, this polynomial is also prime.
3. Polynomial: [tex]\(4x^3 + 2x^2 + 6x + 3\)[/tex]
- If we attempt to factor this polynomial, we find that it does not break down into simpler components with rational coefficients.
- Consequently, this polynomial is prime as well.
4. Polynomial: [tex]\(4x^3 + 4x^2 - 3x - 3\)[/tex]
- This polynomial likewise resists factorization into rational components, indicating it cannot be reduced further over the rationals.
- This means this polynomial is prime.
In conclusion, all four polynomials provided are prime because they are not factorable into polynomials of lower degrees with rational coefficients.
The polynomials that are prime are:
- [tex]\(3x^3 + 3x^2 - 2x - 2\)[/tex]
- [tex]\(3x^3 - 2x^2 + 3x - 4\)[/tex]
- [tex]\(4x^3 + 2x^2 + 6x + 3\)[/tex]
- [tex]\(4x^3 + 4x^2 - 3x - 3\)[/tex]
