How To Factor Expressions In Algebra

In algebra, factoring is one of the most basic methods of simplifying a quadratic equation or expression. Teachers and textbooks often emphasize its importance in basic algebra classes, and with good reason: as students delve deeper and deeper into algebra, they will eventually find themselves dealing with several quadratic expressions at the same time, and factoring helps to simplify them. Once simplified, they become much easier to solve.

1. Find the Key Number for Factoring

Find the key number for the expression by multiplying the whole numbers in the first and last terms of the expression. For example, in the expression 2x² + x – 6, multiply 2 and -6 to get -12.

2. Identify Factors of the Key Number

Calculate factors of the key number that also add up to the middle term. With the expression given above, you must find two numbers that not only have a product of -12, but also have a sum of 1, since there is only a single term in the middle. In this case, the numbers are -12 and 1, since 4 × -3 = -12 and 4 + (-3) = 1.

3. Create a Factoring Grid

Create a 2 × 2 grid and enter the first and last terms of the expression in the upper left-hand corner and lower right-hand corner, respectively. With the expression given above, the first and last terms are 2x² and -6.

4. Fill In the Rest of Your Grid

Enter the two factors into either of the other two boxes of the grid, including the variable as well. With the expression given above, the factors are 4 and -3, and you would enter them into the other two boxes of the grid as 4x and -3x.

5. Find the Common Factor in the Rows

Find the common factor that the numbers in each of the two rows share. With the expression given above, the numbers in the first row are 2x and -3x, and their common factor is x. In the second row, the numbers are 4x and -6, and their common factor is 2.

6. Find the Common Factor in the Columns

Find the common factor that the numbers in each of the two columns share. With the expression given above, the numbers in the first column are 2x² and -4x, and their common factor is 2x. The numbers in the second column are -3x and -6, and their common factor is -3.

7. Complete the Factoring Process

Complete the factored expression by writing out two expressions based on the common factors you found in the rows and columns. In the example examined above, the rows yielded the common factors of x and 2, so the first expression is (x + 2). Since the columns yielded the common factors of 2x and -3, the second expression is (2x – 3). Thus, the final result is (2x – 3)(x + 2), which is the factored version of the original expression.

You can double-check your newly factored expression by multiplying the factor terms together using the FOIL order. That stands for first terms, outer terms, inner terms and last terms. If you've done the math correctly, the result of your FOIL multiplication should be the original, unfactored expression you started with.

You can also double-check your factoring by entering the original expression in a polynomial calculator (see Resources), which will return a set of factors that you can double-check against the result of your own calculations. But keep in mind: Although this type of calculator is useful for quick spot-checks, it's no substitute for learning how to factor algebraic expressions yourself.