factor pair

A factor pair is simply two numbers that multiply together to give a particular target number. The factor pairs of 12, for example, are 1 and 12, 2 and 6, and 3 and 4. When you factor a trinomial, you sift through factor pairs of some target, looking for the one pair that also satisfies a second condition.

In trinomial factoring the target is usually the constant term (for a leading coefficient of 1) or the product a·c (in the AC method). You list the factor pairs of that target and check which pair adds up — or, with the right signs, combines — to the middle coefficient. For x^2 + 7x + 12, the factor pairs of 12 are (1,12), (2,6), (3,4); only (3,4) sums to 7, so those are the numbers you want.

Do not forget negatives. The factor pairs of a positive number include pairs of two negatives (since negative times negative is positive), and the factor pairs of a negative number consist of one positive and one negative. Tracking signs carefully is where most factor-pair searches go wrong.