In the previous investigation, you probably noticed that factoring a trinomial of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi»c«/mi»«/math» is more complex than factoring one 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»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi»c«/mi»«/math». Do not be discouraged. Did you discover the relationships a = mn and c = pq? If you looked really closely, you may have noticed that b = pn + mq. Take a moment to look back at the table in the investigation and confirm, for yourself, that these relationships are true for the examples provided.

Noticing patterns and relationships is a big part of mathematics. The more relationships you see or come to understand, the more connected math concepts will seem. Even if you didn't notice the relationships a = mn, c = pq, and b = pn + mq on your own, taking the time to understand them now will make learning to factor much more manageable because you'll know WHY it works rather than seeing it as 'magic'.

If you are still troubled by the development of these relationships and the factoring method outlined below, please speak with your teacher.

 

These relationships provide enough information to factor a trinomial of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi»c«/mi»«/math», but the procedure is a little less straightforward than for trinomials where a = 1. Can you find a way to use these relationships to factor a trinomial? If you haven't done so already, take a few minutes to try factoring «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»7«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/math» into a product of two binomials of the form .

The relationships discovered in the investigation can be developed symbolically.


Try expanding, and then simplifying, the expression .



The last step was achieved by factoring the GCF, x, from the middle two terms. The result is in the same format as, where , , and . Notice that these relationships are the same as those discovered in the investigation. So, if a trinomial expression can be written in the form , it can be factored. Fortunately, there is an efficient way to do this. Notice that the two values that add to give b, can be multiplied to give the same product as . The conclusion that can be drawn from this is: if you can determine two integers that add to give b and multiply to give ac , the trinomial can be factored.