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Factoring x^6 - y^6 as a difference of squares vs cubes?

  1. Oct 4, 2012 #1
    1. The problem statement, all variables and given/known data

    Factor x6 - y6


    2. Relevant equations

    a3 - b3 = (a - b)(a2 + ab + b2)
    a2 - b2 = (a + b)(a - b)

    3. The attempt at a solution

    I'm confused.

    x6 - y6 = (x2)3 - (y2)3 = (x3)2 - (y3)2

    So shouldn't they all have the same factors? When I factor (x2)3 - (y2)3 = (x3)2 - (y3)2 I get different results.

    (x3)2 - (y3)2
    (x3 - y3)(x3 + y3)
    (x - y)(x2 + xy + y2)(x + y)(x2 - xy + y2)


    (x2)3 - (y2)3
    (x2 - y2)((x2)2 + x2y2 + (y2)2)
    (x – y)(x + y)(x4 + x2y2 + y4)


    How do you factor (x4 + x2y2 + y4) into (x2 + xy + y2)(x2 - xy + y2)? Online factorers are saying it's not possible.
     
  2. jcsd
  3. Oct 4, 2012 #2

    ehild

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    x4 + x2y2 + y4= (x4 + 2x2y2 + y4)-x2y2

    ehild
     
  4. Oct 4, 2012 #3

    Mentallic

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    A little something I also discovered for myself back in high school. I assumed it was a closely guarded secret that only exceptional Mathematicians discover in their lifetimes :wink:
     
  5. Oct 4, 2012 #4

    ehild

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    You are an exceptional Mathematician then... But our teacher told us in the school that adding and subtracting the same thing does not hurt.:smile:

    ehild
     
  6. Oct 4, 2012 #5
    Well, if you work out the product (x^2 + xy + y^2)(x^2 -xy + y^2) you get the right result. What else do you want?
     
  7. Oct 4, 2012 #6

    ehild

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    To find out how to do the factoring if you do not know that these are the factors.

    As I wrote before, x4 + x2y2 + y4= (x4 + 2x2y2 + y4)-x2y2=(x2+y2)2-(xy)2.
    Apply the identity a2-b2=(a-b)(a+b)


    ehild
     
  8. Oct 4, 2012 #7

    Ray Vickson

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    What online factorers are you using? Certainly Maple can do it with no problem (although it is not an on-line package_.

    RGV
     
  9. Oct 4, 2012 #8
  10. Oct 4, 2012 #9

    LCKurtz

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    Last edited: Oct 4, 2012
  11. Oct 4, 2012 #10
    I understand now, thanks for the help :smile:

    Not gonna trust that website as much anymore
     
  12. Oct 4, 2012 #11

    rcgldr

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    What ehild has shown is the difference of two squares, not factors. Note the problem statement is asking for the difference of two squares.
     
    Last edited: Oct 4, 2012
  13. Oct 4, 2012 #12

    Mentallic

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    You should try http://www.wolframalpha.com/
    It can handle much more advanced problems.
     
  14. Oct 4, 2012 #13

    D H

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  15. Oct 4, 2012 #14

    ehild

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    @rcgldr
    I must not give full solution. It was a hint. The difference of two squares is easy to factorize.

    hild
     
  16. Oct 5, 2012 #15

    rcgldr

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    I somehow missed post #6 where you explained this.
     
    Last edited: Oct 5, 2012
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