Factoring: Trinomials

Hello mathematicians, puzzlers, and other curious people!

A lot of algebra students feel like factoring trinomials is a mysterious magic trick. But, there’s really no reason it has to feel this way. Factoring trinomials is really just a strategically selected combination of things you already know how to do — GCF factoring, finding factor pairs for whole numbers, and basic addition or subtraction. (If your memory of any of those is sketchy, it’d be a good idea to review them before going on.)

So, what could you do if you’re told to fully factor $6x^2-21x-12$? (Remember, fully factor means to break it down as far as possible.)

What happens if you don\’t find a factored form that works? If you’ve been systematic and thorough, you probably have a prime trinomial on your hands. You probably remember that prime numbers, like 7 and 23, cannot be broken down into two smaller whole-number factors, and prime polynomials are the same way — they can’t be rewritten as a multiplication problem using smaller polynomial factors.

Figuring out shortcuts to tell whether a trinomial, or other polynomial, is prime would be a good investigation problem. The comments thread is open for that sort of thing, or any other on-topic conversation you’d like to have.