How to Succeed in Algebra - Factoring Non-Monic Trinomials - Part I

One of the stumbling blocks for students in algebra is factoring what is known as non-monic trinomials. Simply stated a non-monic trinomial is one of the form ax^2 + bx + c, in which a is greater than 1.
These present a challenge in factoring because the traditional "guess-and-check" method produces too many possibilities to try.
The approach outlined in this series of articles will not only show the fail-proof way of slaying these trinomials but also give the reason, or proof, as to why this method is valid.

When first learning how to factor trinomials, algebra students are taught the FOIL method, an acronym for First, Outer, Inner, Last: these terms refer to the order in which the expression is multiplied. With an understanding of the FOIL, students then learn essentially how to "reverse FOIL" to get the factored expression from the multiplied trinomial. For example, students first learn to perform (x + 1)*(x + 3) to get x^2 + x + 3x + 3 = x^2 + 4x + 3. Here the First of FOIL refers to the product of x and x; the Outer refers to x*3, the Inner, to 1*x, and the Last to 1*3. The middle term of x and 3x are then combined to give 4x. Once this method is mastered, students then go on to reverse the FOIL so that the trinomial x^2 + 4x + 3 can be factored to (x + 1)*(x + 3).
When the coefficient of the x^2 term is 1, the reverse FOIL is not difficult to master; however, when the coefficient is greater than 1, we end up with a non-monic trinomial (non-monic meaning "not 1), and these present an array of problems for the uninitiated. For example, take the non-monic 6x^2 + 17x + 5.

Since the coefficient of the x^2 term is 6, when reverse FOILing, you have to concern yourself with not x^x but with 6x*x or x*6x, or 2x*3x or 3x*2x to get the first term of the trinomial back. Notice that even though 2x*3x and 3x*2x produce the same result, the order does matter because those factors are then multiplied by the other terms to complete the trinomial. If the order is wrong, the wrong trinomial is obtained. To be specific (3x + 1)*(2x + 5) is not the same as (2x + 1)*(3x + 5).
Do the FOIL on these two to see this.
What is more, if we produce a non-monic which has a large composite leading coefficient and a large composite constant term, the number of possible factors increases enormously, and without a fail-proof system, we may as well give up all hope of factoring the non-monic. Thus trinomials like 18x^2 + 143x + 105 defy easy factorization. For this reason, a sure-fire way to handle such non-monics becomes a welcome guest to any algebra student's toolkit. In Part II of this article, we are going to look at this sure-fire method as well as the proof behind it. After learning this, the algebra student will no longer feel daunted when faced with the need to factor any non-monic regardless of its makeup.
Stay tuned.

.

.