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Trinomial Expansion

  1. Feb 1, 2007 #1
    I find it difficult to expand a trinomial using the formula method (factorial method) where you can find the coefficient of any term without expanding the whole trinomial.
    I can understand the binomial, but I can't do the trinomial using the general sigma notation method.
    Can someone please show me how to do this by using about 2 examples?

    Thanks alot:smile:
     
  2. jcsd
  3. Feb 3, 2007 #2
    anyone can help me? :(
     
  4. Feb 3, 2007 #3

    arildno

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    Let your numbers be a,b,c. Define d=b+c. Then, we have:
    [tex](a+b+c)^{N}=(a+d)^{N}=\sum_{i=0}^{N}\binom{N}{i}a^{(N-i)}d^{i}=\sum_{i=0}^{N}\sum_{k=0}^{i}\binom{N}{i}\binom{i}{k}a^{(N-i)}b^{i-k}c^{k}[/tex]

    Denote the powers of a,b,c as [itex]p_{a},p_{b},p_{c}[/itex], respectively.

    We therefore have that N,i and k are given by:
    [tex]k=p_{c},i=p_{b}+p_{c},N=p_{a}+p_{b}+p_{c}[/tex]
    Thus, your coefficient, in terms of 3 powers are:
    [tex]\binom{p_{a}+p_{b}+p_{c}}{p_{b}+p_{c}}\binom{p_{b}+p_{c}}{p_{c}}[/tex]

    seeing this pattern should tell you how to find the coefficients for higher nomials.
     
    Last edited: Feb 3, 2007
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