← Chapter 6
College Algebra · Chapter 6

The Factor-Pair Hunt

To factor a trinomial you need two numbers — one product, one sum. There is no trick and no guessing required: you list the candidates and test them. Hunt the pairs below, then watch the correct split fall into place, tile by tile.

The box

Why the pair must multiply to a·c

Every tile in the box is (a piece of the top edge) × (a piece of the left edge). So multiply the two corner tiles and you use all four edge pieces — and multiply the two middle tiles and you use those same four edge pieces, just paired up the other way.
corner × corner = (ax2)(c) = ac x2
middle × middle = (mx)(nx) = mn x2
Same four pieces, same product. That forces m·n = a·c. And since the two middle tiles are what add to the middle term, m + n = b. That pair of conditions is the whole method — not a rule to memorize, just the geometry of a rectangle.
The factor-pair huntpick a trinomial to begin
product must be—
sum must be—

The work, one line at a time