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Homework Help: Binomial equations

  1. Jul 21, 2007 #1
    1. The problem statement, all variables and given/known data
    My book says that one "easily" verifies that

    (x+y)^n = (x + y)^(n-2)Q+(x+y)^(n-3)P where

    Q = x^2 + xy +y^2

    and

    P = xy^2 + x^2y


    2. Relevant equations



    3. The attempt at a solution

    I began by expanding everything into summations with binomial coefficients and it seemed like that method would work but it seemed rather far from easy.
     
  2. jcsd
  3. Jul 21, 2007 #2

    Hurkyl

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    Staff Emeritus
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    Try factoring.
     
  4. Jul 21, 2007 #3

    olgranpappy

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    Homework Helper

    They always say "easy" when what they mean that it can be done with a relatively small amount of work. I.e., you need no real inspiration... but it's not necessarily "easy", especially if someone tells you it's "easy." That usually just makes it "frustrating." I hate it when authors use that word. Anyways:
    [tex]
    (x^2+xy+y^2)*(x+y)^{(n-2)}
    +(xy^2+x^2y)*(x+y)^{(n-3)}
    [/tex]
    rewrite the (x+y)^(n-2) in the first term as (x+y)*(x+y)^(n-3) and then factor out the (x+y)^(n-3). you get
    [tex]
    (x+y)^{(n-3)}*\left[((x^2+xy+y^2)*(x+y))+xy^2+x^2y\right]
    [/tex]
    now it should be "easy" to show that the factor in the square bracket is just
    (x+y)^3. So, we are done.
     
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