Question

If you were an Algebra 1 instructor and were creating a test on factoring trinomials of the form ax2 + bx + c, what do you think would be the easiest way to create a trinomial that can be factored? Provide one unique example.

Answer 1

Start with the factors and find the trinomial for it, perhaps.

(x-3)(x+4) = x^2+x-12

Answer 2

Actually, any quadratic of the form ax² + bx + c, where a, b, and c are integers, can be factored if imaginary roots are allowed. However, I am assuming your question pertains to when the quadratic can be factored into rational roots. For this to be true, b² - 4ac must be a perfect square.

Example: in the quadratic x² - 5x + 4, a = 1, b = -5 and c = 4. b² - 4ac = 25 - 4 * 1 * 4 = 25 - 16 = 9, which is a perfect square. And, this quadratic can be factored as (x-4)(x-1).

Furthermore, if b² - 4ac = 0, then the quadratic has a double root. For example, in x² - 6x + 9, b² - 4ac = 36 - 4 * 1 * 9 = 0, and this quadratic can be factored as (x-3)².

Just to be thorough:

If b² - 4ac > 0 but not a perfect square, then it cannot be factored with rational roots -- irrational roots will be required.

If b² - 4ac < 0, then you will need imaginary numbers to factor it.