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Math Help - How to solve Difference of Cubes?

  1. #1
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    How to solve Difference of Cubes?

    Hi, can someone show me how to solve the expression x^6 - y^6?

    It must be done 2 ways: A) Treating it as a different of squares and B) Treating it as a difference of cubes

    I'm really confused. Thanks for help!!
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  2. #2
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    Quote Originally Posted by MATHDUDE2 View Post
    Hi, can someone show me how to solve the expression x^6 - y^6?

    It must be done 2 ways: A) Treating it as a different of squares and B) Treating it as a difference of cubes

    I'm really confused. Thanks for help!!
    A) x^6 - y^6 = (x^3)^2 - (y^3)^2.

    B) x^6 - y^6 = (x^2)^3 - (y^2)^3.

    And I assume you're familar with the formulae for the difference of two squares and for the difference of two cubes.
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    Quote Originally Posted by mr fantastic View Post
    A) x^6 - y^6 = (x^3)^2 - (y^3)^2.

    B) x^6 - y^6 = (x^2)^3 - (y^2)^3.

    And I assume you're familar with the formulae for the difference of two squares and for the difference of two cubes.
    Thanks for the response. And yes, I am familiar with the basic A-B formula. However I don't understand why it expands to become:

    A) (x-y)(x^2+xy+y^2)(x+y)(x^2-xy+y^2)
    B) (x-y)(x+y)(x^4+x^2y^2+y^4) --> if this is a difference of squares, why is there a (x+y)?

    *Answers from the back of the book
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    Quote Originally Posted by MATHDUDE2 View Post
    Thanks for the response. And yes, I am familiar with the basic A-B formula. However I don't understand why it expands to become:

    A) (x-y)(x^2+xy+y^2)(x+y)(x^2-xy+y^2)
    B) (x-y)(x+y)(x^4+x^2y^2+y^4) --> if this is a difference of squares, why is there a (x+y)?

    *Answers from the back of the book
    Please show all your working and say where you get stuck.
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    Quote Originally Posted by mr fantastic View Post
    Please show all your working and say where you get stuck.
    For A) I understand why would factor (x-y) and (x+y) but where is the (x^2+xy+y^2)(x^2-xy+y^2) coming from?


    For B) I know where the (x-y) is coming from (difference of cubes, [a-b]) but where is the (x+y) and (x^4+x^2y^2+y^4) coming from? I kind of understand the trinomial part given that [a^2+ab+b^2]. But other than that, I'm pretty much lost.
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    Quote Originally Posted by MATHDUDE2 View Post
    For A) I understand why would factor (x-y) and (x+y) but where is the (x^2+xy+y^2)(x^2-xy+y^2) coming from?


    For B) I know where the (x-y) is coming from (difference of cubes, [a-b]) but where is the (x+y) and (x^4+x^2y^2+y^4) coming from? I kind of understand the trinomial part given that [a^2+ab+b^2]. But other than that, I'm pretty much lost.
    A^3 - B^3 = .... and A^3 + B^3 = .... .

    I say again, show all your working. (I've asked this for a very good reason so please do it if you want help).
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