Problems with Factoring Binomials

• In math, a "binomial" term involves two different types of numbers, be they variables such as "x" and "y," or a real number and a variable. One method of simplifying binomials to make them easier to understand is "factoring," wherein you represent the binomial in terms of which values you can plug into it to yield zero. Although factoring is a simple and effective process, it is not foolproof.

Difference of Squares

• The correct way to factor binomials is as a "difference of squares." For the binomial a^2 - b^2, the general form for factoring is (a + b)(a - b). You can prove this by multiplying out the factors using the "FOIL"--first, outside, inside, last--method. Multiplying the first terms together gives you a^2, while multiplying the outside yields -ab. Inside terms combine to form ab, while the product of the last two terms is -b^2. After arranging these terms in order--a^2 - ab + ab - b^2--you see that the middle two cancel out, leaving a^2 - b^2.

Square of Differences

• As binomial terms are always the difference between two squares, it's tempting to think of them in terms of the converse--a square of differences. For example, consider the binomial x^2 - 25. Another way to look at this is x^2 - 5^2. Using this logic, you might assume that you can factor the binomial as (x - 5)^2, since both x and 5 are squared to form the binomial. Multiplying this out as before, however, (x - 5)(x - 5) actually yields x^2 - 10x + 25, a very different value than the actual binomial.

Coefficients

• Another potential source of confusion when factoring binomials might be the presence of a "coefficient." For example, if you have the binomial 4x^2 - 36, you might ignore the fact that 4 is the square of 2 and factor your binomial as (4x - 6)(x + 6). Multiplying this out using FOIL, however, you would find that you have actually factored the expression 4x^2 + 18x - 36. As is the case with ordinary variables, you must factor only the squares of variables with coefficients. Another example of this is 25x^2 - 144--you must factor this as (5x - 12)(5x + 12).