Question

Which polynomials are prime? Check all of the boxes that apply.

x2 + 9

x2 – 9

x2 + 3x + 9

–2x2 + 8

Answer 1

Assuming “prime” means irreducible over the integers (or rationals):

- x^2 + 9 — prime. Discriminant = 0^2 − 4·1·9 = −36 < 0, so no rational roots and it does not factor over Z or Q.
- x^2 − 9 — not prime. x^2 − 9 = (x − 3)(x + 3).
- x^2 + 3x + 9 — prime. Discriminant = 3^2 − 4·1·9 = −27 < 0, so it has no rational factorization.
- −2x^2 + 8 — not prime. −2x^2 + 8 = −2(x^2 − 4) = −2(x − 2)(x + 2).

So check: x^2 + 9 and x^2 + 3x + 9.