How to Factor Prime Trinomials

By Maria O'Brien
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If you are asked to factor a prime trinomial, do not despair. The answer is quite easy. Either the problem is a typo or a trick question: by definition, prime trinomials can not be factored. A trinomial is an algebraic expression of three terms, for instance x2 + 5 x + 6. Such a trinomial can be factored--that is, expressed as the product of two or more polynomials. This example can be factored into (x + 3) (x + 2). Notice that the trinomial was of second degree (second power), but the binomial factors were of first degree. A prime trinomial can not be written as the product of lower degree polynomials. How can you tell if you have a prime trinomial? Read on to find the answer.

Write the factors of the constant term, if the trinomial is of the form x2 + bx + c. In this form, c is the constant and the coefficient of the x2 term is 1.

Note that If any of the factor pairs of c add up to b, the trinomial is not prime. In the example above, the factors of the constant 6, are 1 * 6 and 2 * 3 (also -1 * -6 and -2 * -3). Because the factor pair 2 and 3 add up to 5, you know that this trinomial can be factored and is NOT prime.

Look at it from another angle. On the other hand, for the trinomial x2 - 11x - 10, the factor pairs for the constant ( - 10) are -1 * 10; -2 * 5, -5 * 2 and -10 * 1. The sums of these factors are, respectively, -9, 3, -3 and -9. None of these sums is equal to the coefficient of the x term, -11. Therefore, this is a prime trinomial.