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Math Help - Special Factoring Formulas

  1. #1
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    Special Factoring Formulas

    Does anyone know how to derive the factor formulas for sum and difference of cubes? that is
    <br />
a^3 + b^3 = (a + b) (a^2 - ab + b^2)<br />
    <br />
a^3 - b^3 = (a - b) (a^2 + ab + b^2)<br />
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  2. #2
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    Quote Originally Posted by skylabel View Post
    Does anyone know how to derive the factor formulas for sum and difference of cubes? that is
    <br />
a^3 + b^3 = (a + b) (a^2 - ab + b^2)<br />
    <br />
a^3 - b^3 = (a - b) (a^2 + ab + b^2)<br />
    a^3 + b^3

    a^3  -a^2b+ab^2+ ba^2 - ab^2 + b^3

    a^3+ ba^2  - a^2b - ab^2 +ab^2+ b^3

    a^2(a+b)  - ab(a+b) +b^2(a+b)

    (a^2 - ab + b^2)(a+b)
    .................................................. ...........................
    <br />
a^3 - b^3

    a^3  -a^2b+a^2b+ b^2a - ab^2 - b^3

    a^3- ba^2  + a^2b - ab^2 +ab^2- b^3

    a^2(a+b)  + ab(a-b) +b^2(a-b)

    (a^2 + ab + b^2)(a-b)
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    I have just multiplied the factors to show you the steps
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  3. #3
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    Quote Originally Posted by skylabel View Post
    Does anyone know how to derive the factor formulas for sum and difference of cubes?
    Another method is to do the polynomial long division:

    [HTML] a^2 + ab + b^2
    --------------------------
    a - b )a^3 + 0a^2b + 0ab^2 - b^3
    a^3 - 1a^2b
    --------------------------
    1a^2b + 0ab^2 - b^3
    1a^2b - 1ab^2
    --------------------------
    1ab^2 - b^3
    1ab^2 - b^3
    --------------------------
    0[/HTML]
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  4. #4
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    Quote Originally Posted by stapel View Post
    Another method is to do the polynomial long division:

    [html] a^2 + ab + b^2
    --------------------------
    a - b )a^3 + 0a^2b + 0ab^2 - b^3
    a^3 - 1a^2b
    --------------------------
    1a^2b + 0ab^2 - b^3
    1a^2b - 1ab^2
    --------------------------
    1ab^2 - b^3
    1ab^2 - b^3
    --------------------------
    0[/html]
    To elaborate:

    Since a^3 - b^3 is equal to zero when a = b, it's obvious that a - b is a factor of a^3 - b^3.
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