Looking at a number of specific examples, like you did in the Investigation, is a good way to look for patterns and relationships. Another strategy for determining relationships is to use a more general version of the scenario. The following explanation shows a relationship between p, q, b, and c, and outlines how this relationship can be used to factor some trinomials. Make note of how this relationship compares to the one you observed in the Investigation.

Factoring and expanding are opposite processes, so start by imagining the expansion of two factors and then look to see how you could work backwards from the expansion to determine the factors.


Suppose a trinomial can be factored into .

Multiplying these factors gives

Notice that p + q is the b-value and pq is the c-value of a trinomial of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi mathcolor=¨#E5C304¨»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi mathcolor=¨#990000¨»c«/mi»«/math». This helpful result means that if there are two numbers, p and q, that add to give b and multiply to give c, the trinomial can be factored as .

Key Lesson Marker


Both the pattern method and the algebraic method of determining the relationship between p, q, b, and c lead to the same strategy for factoring a trinomial of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#191919¨»«mi mathcolor=¨#191919¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#191919¨»+«/mo»«mi mathcolor=¨#191919¨»b«/mi»«mi mathcolor=¨#191919¨»x«/mi»«mo mathcolor=¨#191919¨»+«/mo»«mi mathcolor=¨#191919¨»c«/mi»«/math».

  • Determine two integers, p and q, that add to give b and multiply to give c.

  • Write the factors as .

  • Verify by multiplying the factors.