A Better Way to Factor Trinomials!

Yes, that’s really the topic of this post. Maybe only we mathematicians get excited about such things, but I thought this was so neat when I saw it in L’s math book. She is using an algebra book from the series called The Life of Fred. It has worked well for my social, verbal daughter, because the math is taught in the context of a story. What I like, though, is that Fred has some unique ways of doing things.

Remember (or maybe you don’t) how it’s easy to factor trinomials when the coefficient of the x² term is one? But it gets a lot more complicated when that number is something bigger. I was taught basically to make successive guesses until I found what works. But Fred showed me this way--

Factor 3x² + 11x + 6

Consider 3 (the coefficient of x²) and 6 (the constant term).

Which of their factors, when recombined into 2 new numbers, will add to 11 (the coefficient of x)?

3=1*3

6=2*3 or 1*6

Recombining those factors, I get 1*2=2 and 3*3=9. The sum of 2 and 9 is 11.

(I didn’t use the factors 1*6 because they don’t combine with the factors of 3 to get 11.)

Now rewrite the problem: 3x² + 9x + 2x + 6.

(This is exactly the same as above, but the 11x is split into 9x + 2x.)

Find the common factor in the first two terms, then in the second two:

3x(x + 3) + 2(x + 3)

Find the common factor again. It is (x + 3):

(x + 3)(3x + 2)

Done! You have to factor twice, but it’s much easier than the guessing game I learned in my algebra classes.

This algebra book is the only one I have used from the Life of Fred. As I mentioned, it has been a good fit for my daughter. It is the only math series I know of that is "whole to parts." I feel, however, that there are some places where Fred moves too quickly. There have been a few times when I have supplemented with additional explanation. I also highly recommend the Fred Companion book for this same reason; it helps to pace one’s progression through the chapters, and it offers more problems for extra practice.