Question

Which polynomial is prime?

3x3 + 3x2 – 2x – 2
3x3 – 2x2 + 3x – 4
4x3 + 2x2 + 6x + 3
4x3 + 4x2 – 3x – 3

Answer 1

To determine which polynomial is prime (meaning it cannot be factored into the product of two non-constant polynomials with coefficients in the same field, which in this case is likely the field of rational numbers), we can try to factor each polynomial or check their properties.

Let's analyze each polynomial:

1. **\(3x^3 + 3x^2 - 2x - 2\)**:
- We can try factoring by grouping or using the Rational Root Theorem.
- Using synthetic or polynomial division can also offer insights.
- Upon inspection or application of roots, we find that it factors as \((3x + 2)(x^2 + x - 1)\).

2. **\(3x^3 - 2x^2 + 3x - 4\)**:
- Similar techniques can be used.
- This polynomial does not have simple roots and does not factor nicely.
- It's necessary to check if it can be expressed as a product of two polynomials, which typically requires further investigation.

3. **\(4x^3 + 2x^2 + 6x + 3\)**:
- Again, we can try to find rational roots or utilize polynomial division.
- It appears to factor as \((2x + 3)(2x^2 + 1)\), hence it is not prime.

4. **\(4x^3 + 4x^2 - 3x - 3\)**:
- For this polynomial, we can also check for factors or rational roots.
- It turns out it factors to \((4x + 3)(x^2 - 1)\), thus also not prime.

After analyzing each polynomial, it appears that the second polynomial, **\(3x^3 - 2x^2 + 3x - 4\)**, may indeed be prime as it does not yield obvious factors or roots with rational coefficients.

Thus, the polynomial that is likely to be prime is:
**\(3x^3 - 2x^2 + 3x - 4\)**