## Using Polynomial Multiplication To Foreshadow Factoring

In today’s elementary algebra class we started to cover factoring by going over how to factor the GCF from a polynomial. Once we figured out how to find the GCF, I told my students that they needed to divide this factor from each term, or “undistribute” the GCF from the polynomial. It was at this point that I had an idea – why didn’t I just teach this when I was teaching my students how to multiply a monomial by a polynomial? The reason is a practical one – it would stretch out the polynomial chapter by at least a week. But that doesn’t mean we cannot incorporate this idea to plant a seed for later.

For example, after your students learn to multiply a monomial by a polynomial, ask them to find a polynomial that satisfies

2x ( ? ) = 8x^5 – 4x^3 + 10x

(?) (x^3 + 5x^2 – 7) = 9x^7 + 45x^6 -63x^4

These are the intuitive skills that students need for factoring out the GCF. When you teach your students how to multiply binomials (with or without FOIL), why not foreshadow factoring trinomials?

(x + ?) (x + ?) = x^2 + 13x +40
(x – ?) (x + ?) = x^2 -3x – 54
(2x + 7) (?) = 6x^2 +31x + 35

The same foreshadowing can be done for difference of squares, maybe even for difference of cubes or sum of cubes.

Foreshadowing in this manner helps your students to develop a deeper intuitive understanding of the current material, as well as preparing them for future topics. This approach can be used for any topic that leads to future topics, and we know that mathematics is full of opportunities in this way.

I’d love to hear what you think of my idea. I’d REALLY love to hear whether you use this strategy, and for which topics. I encourage you to share by leaving a comment, or reaching me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to general teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

• 1. David  |  April 14, 2010 at 10:07 pm

Great idea! I am already well into factoring this semester, but I will follow your suggestion the next time I teach polynomial multiplication.

This semester, I used factoring by grouping as my first method for factoring quadratics. For example:

x^2 + 3x + 2 = x^2 + x + 2x + 2 = (x^2 + x) + (2x + 2) = x(x+1) + 2(x+1) = (x+2)(x+1).

I pointed out that checking the solution just amounts to reading the steps in reverse order:

(x+2)(x+1) = x(x+1) + 2(x+1) = (x^2+x) + (2x+2) = x^2 + x + 2x + 2 = x^2 + 3x + 2.

There are quicker methods, but they don’t make it as clear that factoring is just multiplying in reverse.

• 2. georgewoodbury  |  April 14, 2010 at 10:48 pm

David, I like what you’re doing there too. With trinomials we need to get away from just finding two magic numbers and making sure students understand the concept. It doesn’t take a lot of time or effort to bring these ideas up in class, whether through a quick pointer or carefully crafted question, but the payoff can be huge.