Find all integers b so that the trinomial 3x^2 +bx + 2 can be factored.

Answer 1

The discriminant must be positive for there to be real roots, that is, factors

so, b^2 - 24 > 0
b must be an integer such that |b| >= 5

3x^2-5x+2 = (3x-2)(x-1)
3x^2+5x+2 = (3x+2)(x+1)

3x^2-7x+2 = (3x-1)(x-2)
3x^2+7x+2 = (3x+1)(x+2)

Many others are possible, not all of which factor into rational roots. Any integer b > 5 will provide real roots, however.