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Special Factoring:
     Factoring Differences of Squares (page 1 of 3)

Sections: Differences of squares, Sums and differences of cubes, Recognizing patterns

When you learn to factor quadratics, there are three other formulas that they usually introduce at the same time. The first is the "difference of squares" formula.

Remember from your translation skills that "difference" means "subtraction". So a difference of squares is something that looks like x² – 4. That's because 4 = 2², so you really have x² – 2², a difference of squares. To factor this, do your parentheses, same as usual:

x² – 4 = (x      )(x     )

You need factors of –4 that add up to zero, so use –2 and +2: 

x² – 4 = (x – 2)(x + 2)

(Review Factoring Quadratics, if this example didn't make sense to you.)

Note that we had x² – 2², and ended up with (x – 2)(x + 2). Differences of squares (something squared minus something else squared) always work this way:

For a² – b², do the parentheses:

(         )(        )

...put the first squared thing in front:

(a      )(a       )

...put the second squared thing in back:

(a    b)(a     b)

...and alternate the signs in the middles:

(a – b)(a + b)

Memorize this formula! It will come in handy later, especially when you get to rational expressions (polynomial fractions), and you'll probably be expected to know the formula for your next test.

Here are examples of some typical homework problems:

• Factor x² – 16

This is x² – 4², so I get:

x² – 16 = x² – 4² = (x – 4)(x + 4)

• Factor 4x² – 25

This is (2x)² – 5², so I get:

4x² – 25 = (2x)² – 5² = (2x – 5)(2x + 5)

• Factor 9x⁶ – y⁸

This is (3x³)² – (y⁴)², so I get:

9x⁶ – y⁸ = (3x³)² – (y⁴)² = (3x³ – y⁴)(3x³ + y⁴)

• Factor x⁴ – 1 Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

This is (x²)² – 1², so I get:

x⁴ – 1 = (x²)² – 1² = (x² – 1)(x² + 1)

Note that I'm not done yet, because x² – 1 is itself a difference of squares, so I need to apply the formula again to get the fully-factored form. Since x² – 1 = (x – 1)(x + 1), then:

x⁴ – 1 = (x²)² – 1² = (x² – 1)(x² + 1)

          = ((x)² – (1)²)(x² + 1)

          = (x – 1)(x + 1)(x² + 1)

The answer to this last exercise depended on the fact that 1, to any power at all, is still just 1.

Warning: Never forget that this formula is for the difference of squares (with variables); the polynomial sum of squares is always prime (that is, it can't be factored with whole numbers or fractions).

You can use the Mathway widget below to practice "Factoring Polynomials", subtopic "Differences of Squares". Try the entered exercise, or type in your own exercise. Then click "Answer" to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "View Steps" on the widget's answer screen will take you to the Mathway site, where you can register for a free seven-day trial of the software.)

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