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Math Help - (a + b + c)^3

  1. #1
    Senior Member pacman's Avatar
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    (a + b + c)^3

    Verify this identity: (a + b + c)^3 + (a - b - c)^3 - (a + c - b)^3 - (a + b - c)^3 = 24abc
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  2. #2
    Member rowe's Avatar
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    This is just a matter of expanding each bracketed expression, and then simplifying everything. Tedious, more than anything. Here's the first bracketed term expanded.

    (a + b + c)^3

    =(a+b+c)(a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2)

    =(a+b+c)(a^2 + 2ab + 2ac + b^2 + 2bc + c^2)

    =a^3+3a^2 b+3a^2c+3ab^2+6abc+3ac^2+b^3+3b^2c+3b c^2+c^3
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  3. #3
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    Grandad's Avatar
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    Hello pacman
    Quote Originally Posted by pacman View Post
    Verify this identity: (a + b + c)^3 + (a - b - c)^3 - (a + c - b)^3 - (a + b - c)^3 = 24abc
    First, can I encourage you to use LaTeX: it makes your questions and working much more readable. In this instance, all you need do to change your original expression to LaTeX is to select it, and then press the \Sigma button on the toolbar. Which is what I've done here:

    (a + b + c)^3 + (a - b - c)^3 - (a  + c - b)^3 - (a + b - c)^3  = 24abc

    Then: I think we can reduce the tedium a bit by noting one or two well-known results first:

    p^3 + q^3 = (p+q)(p^2-pq+q^2) (1)

    p^2-q^2 = (p-q)(p+q) (2)

    (p+q)^2+(p-q)^2 = 2p^2+2q^2 (3)

    We begin by re-grouping a little:

    (a+b+c)^3+(a-b-c)^3-(a+c-b)^3 -(a+b-c)^3 =\Big((a+[b+c])^3+(a-[b+c])^3\Big)-\Big((a+[c-b])^3 +(a-[c-b])^3\Big)

    Then, using (1) on each expression in the big brackets:

    = 2a\Big((a+[b+c])^2-(a+[b+c])(a-[b+c]) +(a-[b+c])^2\Big) - 2a\Big((a+[c-b])^2-(a+[c-b])(a-[c-b]) +(a-[c-b])^2\Big)

    =2a\Big(2a^2 +2(b+c)^2 -a^2 +(b+c)^2\Big)-2a\Big(2a^2+2(c-b)^2-a^2+(c-b)^2\Big), using (2) and (3)

    =2a\Big(a^2 +3(b+c)^2\Big)-2a\Big(a^2+3(c-b)^2\Big)

    =2a\Big(3(b+c)^2-3(c-b)^2\Big)

    =6a(b^2+2bc+c^2-c^2+2bc-b^2)

    =24abc

    Grandad
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  4. #4
    Senior Member pacman's Avatar
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    Thanks Grandad, your method is excellent . . . . i have tried same as that of Rowe, it overlapped my paper. . . . .

    And i will learn LateXing . . . .
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