## Factoring Trinomials – Trial and Error or Grouping?

Yesterday I was teaching my students how to factor trinomials with a leading coefficient that is greater than 1, such as 6x^2 – 25x + 24. There are two methods for doing this – “trial and error” and “grouping”. There are strengths and weaknesses to both approaches. In my experience it is wise to select one method and stick with it, but yesterday I showed both techniques.

### Trial and Error

This method, as its name implies, is all about trying possible factors until you find the right one. 6x^2 can be expressed as x(6x) or 2x(3x), so if the trinomial factors it will be of the form (x-?)(6x-?) or (2x-?)(3x-?). Now we replace the question marks by the factor pairs of 24 (1 & 24, 2 & 12, 3 & 8, 4 & 6) in all possible orders until we find the correct pair of factors that produce the “middle term” of -25x. The correct factoring is (2x-3)(3x-8). Check for yourself to be sure ;-).

I like this technique because it helps students develop their mathematical intuition. It is similar to the method we use to factor quadratic trinomials with a leading coefficient of 1. Students can make their work easier by recognizing that the two terms in a binomial factor cannot have a common factor, allowing them to skip certain pairings. For example, (x-1)(6x-24) cannot be correct because 6x and 24 contain a common factor. In the example I gave, there are 16 possible factorizations to check.

14 of the factorizations contain a common factor and can be skipped:

(x-1)(6x-24), (x-2)(6x-12), (x-12)(6x-2), (x-3)(6x-8), (x-8)(6x-3), (x-4)(6x-6), (x-6)(6x-4),
(2x-1)(3x-24), (2x-24)(3x-1), (2x-2)(3x-12), (2x-12)(3x-2), (2x-8)(3x-3), (2x-4)(3x-6), (2x-6)(3x-4)

Only 2 of the factorizations need to be checked: (x-24)(6x-1) and (2x-3)(3x-8)

So, a student can really reduce their workload and factor this trinomial fairly quickly.

Some students don’t like it because there is no definite procedure leading to a solid “answer”. Some students do not like trying, and trying, and trying, until they find the right factors.

### Grouping

Using grouping makes use of the students’ knowledge of FOIL.

To factor 6x^2 – 25x + 24 using grouping, we need to work backwards. In other words, the student must find a way to rewrite -25x as -16x-9x. To determine how to split up the middle term, students multiply the first and last coefficients: 6(24) = 144. Now they need to find two integers that multiply to 144 and add to -25. This part of the problem is also similar to factoring quadratic trinomials with a leading coefficient of 1. The problem is that the numbers students are now working with are larger – it will take students a little while to list the factors of 144 until they realize that -19 and -6 are the two integers we are seeking.

Students like this method because it’s a clear, step-by-step process that will lead to the correct factorization.

### So, Which One Should We Use?

Both methods build on previous techniques and topics, and therefore can be used to help students increase their conceptual understanding. I prefer trial and error because I think it encourages creativity, and helps students to use finesse over “brute force”. Students can use their intuition to focus in on likely correct answers. Students can speed up the process by eliminating impossible factorizations.

But there have been semesters in which I used grouping. It builds upon factoring by grouping in general, as well as FOIL and some of the skills used in factoring trinomials with a leading coefficient of 1.

This is the first time I used both methods in my class. I figured that students could use the method that seemed best to them. About 2/3 of my students preferred “trial and error”, for what it’s worth. I’m not sure if that is because I introduced it first, or whether I’ve developed a classroom of creative/intuitive students.

### How Do You Do It?

Which technique do you use in class, or do you use both? What is the reasoning behind your choice?  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. Andy Hynds  |  April 23, 2010 at 11:54 am

I also have gone in both directions on this one. I often remind my students that there is not one consistent way to do it every single time, but there are some strategies that can lessen the amount of “guessing,” which are:

1) Don’t include anything that leads to common factors
2) Consider the + and – signs
3) Think about even and odd numbers
4) Lean more towards the middle numbers for your first guess – 35 can be 35×1 or 5×7, but 5 and 7 are more likely.

Thank you for the suggestions. I just finished this unit, and I might look at this when I teach the concept again in the fall.

• 2. shana donohue  |  June 18, 2010 at 10:01 am

What a great question! I grew up using trial and error for trinomials with A greater than 1, and it was so frustrating! It wasn’t until (and I’m embarassed admitting this) I took a college algebra class in graduate school that I finally learned that there was a method to the madness. I will be posting a new animated video on my site that shows how to factor trinomials with A greater than 1.

Great question!!

• 3. shana donohue  |  June 18, 2010 at 3:01 pm

Ok, I made the animation on factoring trinomials….

http://zerosumruler.wordpress.com/

I hope you like it (and post suggestions or feedback)!

• 4. georgewoodbury  |  June 18, 2010 at 9:21 pm

Well done Shana!

• 5. shana donohue  |  June 19, 2010 at 5:41 am

Thak you George!

• 6. sherilyn laxamana  |  October 10, 2011 at 5:39 am

. ohh thanks 🙂

• 7. Shana Donohue  |  October 10, 2011 at 6:42 am

I remember factoing trinomials with Non-1 A as a kid and thought it was the messiest thing about math. I wasn’t shown the method, just trial and error. I wanted it to be more methodical like the rest of class.

As a grad student, I was finally shown the method. It’s awesome. Have you seen its proof? Now I teach the method so that my students can find the success I never did. I’d love to start with the proof, however it’s a little much for intro Algebra.

• 8. Rebecca  |  March 26, 2012 at 1:02 am

I would love to see the proof- been trying to figure it out all night! Don’t want to teach method until I understand why it works.

• 9. ZeroSum Ruler  |  March 26, 2012 at 2:11 pm

Hi Rebecca,

I’m teaching my 8th graders how to factor all trinomials now! Here is the animated proof I created from a proof I found (professor credited at end of video). Would love your feedback!