Question

Which polynomial is prime?

x3 + 3x2 + 2x + 6
x3 + 3x2 – 2x – 6
10x2 – 4x + 3x + 6
10x2 – 10x + 6x – 6

Answer 1

Simplify and check factorability:

1) x^3 + 3x^2 + 2x + 6 = x^2(x+3) + 2(x+3) = (x^2+2)(x+3) — reducible.

2) x^3 + 3x^2 − 2x − 6 = x^2(x+3) − 2(x+3) = (x^2−2)(x+3) — reducible.

3) 10x^2 − 4x + 3x + 6 = 10x^2 − x + 6. Its discriminant is b^2 − 4ac = (−1)^2 − 4·10·6 = 1 − 240 = −239, not a perfect square (indeed negative), so it has no rational linear factors — irreducible over the rationals (prime).

4) 10x^2 − 10x + 6x − 6 = (10x^2 −10x) + (6x −6) = 10x(x−1)+6(x−1) = (10x+6)(x−1) — reducible.

Therefore the third polynomial (10x^2 − 4x + 3x + 6, i.e. 10x^2 − x + 6) is prime.