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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:

 

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    For a2b2, 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:

      (ab)(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:

      9x6y8 = (3x3)2 – (y4)2 = (3x3y4)(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 a difference 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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Cite this article as:

Stapel, Elizabeth. "Special Factoring: Differences of Squares." Purplemath. Available from
    http://www.purplemath.com/modules/specfact.htm. Accessed
 

 

 

 

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