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I can't understand a development problem, I need help, please!

  1. Dec 21, 2013 #1
    1. The problem statement, all variables and given/known data

    I'm learning mathematics in Japanese, so maybe my English is not correct to discribe what I want to mention.
    I'm struggling a development probloem below

    ( x + y + 2z)3 – ( y + 2z – x)3 – ( 2z + x – y )3 – ( x + y – 2z )3




    2. Relevant equations

    I used cube development formula like a3 + b3= ( a+ b )3 -3ab( a + b)


    3. The attempt at a solution
    1. The problem statement, all variables and given/known data

    I solved this problem, but the answer of my reference book used another way to solve it. Then, I can't understand that way.

    My attempt at a solution is below

    transpose   {x + y + 2z}3 - { y+2z - x }3=P,
    { 2z + x - y }3 + { x + y - 2z}3=Q

    P=[{( y+2z) + x } - {( y + 2z) - x}]3 + 3{( y + 2z ) + x}{( y + 2Z) - x}{( y + 2z + x) - ( y + 2z - x)}
    =(2x)3 + 3{( y + 2z )2 - x2} ・2x
    =8x3 + 6x( y2 + 4yz + 4z2 - x2)
    =8x3 - 6x3 + 6xy2 + 24xyz + 24z2x
    =2x3 + 6xy2 + 24z2x - 24xyz


    Q=[{ x -( y - 2z)} + { x + ( y - 2z)}]3 - 3{ x - ( y - 2z)}{ x + ( y - 2z)}×( x - y + 2z + x + y -2z)
    =(2x)3 - 3{ x2 - ( y - 2z)2}・2x
    =8x3 - 6x( x2 - y2 + 4yz - 4z2)
    =2x3 - 6xy2 + 24z2x - 24xyz

    thus,

    P-Q=( 2x3 + 6xy2 + 24z2x + 24xyz) - ( 2x3 + 6xy2 + 24z2x - 24xyz)
       =48xyz

    2. Relevant equations

    I solved this problem like above, but the answer of the reference book is different from mine.
    I can't this way to solve this problem. Please help me.

    The answer of reference book is below

    transpose   {x + y + 2z}3 - { y+2z - x }3=P,
    { 2z + x - y }3 + { x + y - 2z}3=Q

    P={( x + y + 2z) - ( y + 2z - x)}{( x + y + 2z)2 + ( x + y +2z )( y + 2z - x) + ( y + 2z - x)2}
    =2x{ x2 + 2x( y + 2z) + ( y + 2z)2 + ( y + 2z)2 - x2 + (y + 2z)2 - 2x( y + 2z) + x2}
    =2x{3( y + 2z)2 + x2}

    Q={(2z + x - y) + ( x + y - 2z)}{(2z + x - y)2 - ( 2z + x - y)( x + y - 2z) + (x + y - 2z)2}
    =2x{(2z - y)2 + 2x(2z - y) + x2 - x2 + (2z - y)2 + (2z - y)2 - 2x( 2z - y) + x2}
    =2x{3(2z - y)2 + x2}

    thus,

    P-Q=2x{3(y + 2z)2 + x2 - 3( 2z - y)2- x2}
    =6x・8yz=48xyz



    the answer is same to my result, but the way to solve this problem is different. I want to understand the solving way of reference book. Please help me.
     
    Last edited by a moderator: Dec 21, 2013
  2. jcsd
  3. Dec 21, 2013 #2

    Mentallic

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    The reference book is using the rule

    [tex]a^3+b^3=(a+b)(a^2+ab+b^2)[/tex]

    and

    [tex]a^3-b^3=(a-b)(a^2-ab+b^2)[/tex]
     
  4. Dec 23, 2013 #3

    haruspex

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    You mean

    [tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]

    and

    [tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex]
     
  5. Dec 23, 2013 #4

    Mentallic

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    Yeah :biggrin:
     
  6. Dec 24, 2013 #5
    I noticed that, but I could find my mistake

    Yes, that formula is true. I found that the first comment has a little mistake. But, I could find my mistake by the comment. I'm grateful for two members, thank you very much.
    I'm poor at English and also mathematics, so if you advise me when I ask for help, I'll be really grateful for that.
     
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