Question

Which polynomial is prime?

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

Answer 1

None — each one factors over the integers:

- 3x^3 + 3x^2 – 2x – 2 = (x + 1)(3x^2 – 2)
- 3x^3 – 2x^2 + 3x – 4 = (x – 1)(3x^2 + x + 4)
- 4x^3 + 2x^2 + 6x + 3 = (2x + 1)(2x^2 + 3)
- 4x^3 + 4x^2 – 3x – 3 = (x + 1)(4x^2 – 3)

So none of the given cubics is prime (irreducible) in Z[x] (or Q[x]).