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Factoring A Difference Between Two Squares Lessons

Take a look at the problem and work below:

(x + 3)(x - 2)
x2 - 2x + 3x -6
x2 + x - 6

As you can see above, two binomials in parentheses, (x + 3) and (x - 2) are multiplied using the FOIL method. When the outer terms and inner terms are multiplied they result in -2x and 3x. Because these two resulting terms have different coefficients, when we combine like terms the result is another term, +x, which appears in the last line of the problem.

Now observe the following problem:

(x - 2)(x + 2)
x2 + 2x - 2x - 4
x2 - 4

In this case, the multiplication of the outer terms and of the inner terms resulted in 2x and -2x. The terms 2x and -2x have coefficients of 2 and -2. These coefficients are essentially the same number, but with with opposite signs (one number is positive and the other is negative). These two terms form a zero pair, meaning when they are combined, they cancel each other out. You can see that in the last step of the problem, where like terms were combined, the zero pair 2x and -2x canceled out. As a result, only two terms remained on the last line.

The result from the last problem is called a Difference Between Two Squares. A Difference Between Two Squares is an expression with two terms (also known as a binomial) in which both terms are perfect squares and one of the two terms is negative.

The pages that follow show how to factor a difference between two squares. The factoring process, which converts an expression like "x2 - 4" into "(x - 2)(x + 2)", is essentially the opposite of the multiplication process we used above.


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