The Box Method for Factoring

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this lesson, you'll know how to use the box method to help you factor polynomials that may not be as easy to factor using other methods. You'll see how the box method breaks down the factoring for you.

A Quadratic Polynomial

In this lesson, you'll learn another method that can help you factor quadratic polynomials. A polynomial is a mathematical equation with terms made up of coefficients and variables. A quadratic polynomial is a polynomial where the highest exponent is 2. These are examples of quadratic polynomials:

box method

All of your quadratic polynomials will follow this general form:

box method

The a, b, and c are the coefficients and x is your variable.

In math class, you'll need to be able to factor these quadratic polynomials so you can find their solutions. Well, in this lesson, you'll learn about the box method of factoring. This method actually makes your job a bit easier. It keeps your factored terms separate and provides a great visual to show you just what your factors will be.

Let's see what this method is all about. We'll talk about the method briefly and then we'll dive into using the method to factor one of the above polynomials.

The Box Method

The box method, as its name implies, uses a box to help you factor your quadratic polynomials. The box is actually a square divided into four equal parts to help you separate your quadratic into its factors. Now, do keep in mind that you can't factor all quadratics. Some just don't have any real factors. But for those quadratic polynomials that do have factors, you can use this box method to make the process easier.

It begins with you drawing a 2 x 2 grid, or a square divided into four equal parts. Then you write your first squared term in the upper left box and your last term in the lower right box. You then multiply your first term with your last term. You'll find the factors of this new term. Specifically, you are looking for a set of factors that when you add them you get your middle term. Once you've found this set of factors, you'll write them in the two remaining blank boxes. It doesn't matter in which empty box you write each of the terms, your answer will still be the same. Now, to find the factors of your quadratics, you'll find the greatest common factor of each row and column. Then the resulting greatest common factors you get from your columns and rows respectively will be the factors of your polynomial.

Now that we've described the box method, let's take a look at it in action.

Using the Box Method

Let's use the box method to solve our first quadratic, 6x^2 - x - 12.

We begin by drawing a 2 x 2 grid.

box method

We then write the first term in the upper left box and the last term in the lower right box.

box method

Next, we multiply our first term with our last term. Remember to keep any negative signs you see.

box method

Now, we need to find a set of factors of -72x^2 that when added together will give us our middle term, -x. Separating the -72x^2 into its factors, we get 12x and -6x, -12x and 6x, 8x and -9x, -8x and 9x, 2x and -36x, -2x and 36x, 3x and -24x, -3x and 24x, 4x and -18x, -4x and 18x, x and -72x, and finally -x and 72x. Looking at all these possibilities, the only one that will give a -x when added together is the 8x and -9x combination.

We'll put these two factors in the remaining boxes. It doesn't matter which goes in what box as the answer will still be the same.

box method

Now, we go ahead and find the greatest common factor for each row and each column. So, we'll be finding the greatest common factor between 6x^2 and 8x, between -9x and -12, between 6x^2 and -9x, and between 8x and -12. We'll write each of these greatest common factors next to each row and column respectively.

box method

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