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Math Help - Binomial Theorem with coefficients

  1. #1
    Senior Member Paze's Avatar
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    Binomial Theorem with coefficients

    I can get the binomial theorem to work just fine when doing (a+b)^n but it becomes a problem as soon as I put coefficients, e.g.: (2a+3b)^n

    Why? What is the difference and how do I calculate with coefficients?
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  2. #2
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    Re: Binomial Theorem with coefficients

    Well, there's no difference.

    (a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k

    (2a+3b)^n = \sum_{k=0}^{n}\binom{n}{k}(2a)^{n-k}(3b)^k=\sum_{k=0}^{n}\binom{n}{k}2^{n-k}a^{n-k}\cdot 3^kb^k

    Um, did this help?
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  3. #3
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    Re: Binomial Theorem with coefficients

    Hello, Paze!

    I can get the binomial theorem to work just fine when doing (a+b)^n
    but it becomes a problem as soon as I put coefficients, e.g.: (2a+3b)^n

    Why? What is the difference and how do I calculate with coefficients?

    Consider n=3\!:\;(x+y)^3 \:=\:x^3 + 3x^2y + 3xy^2 + y^3 .[1]

    Suppose we are given: . (2a + 3b)^3

    Then we have: . \begin{Bmatrix}x \:=\: 2a \\ y \:=\: 3b\end{Bmatrix}


    Substitute into [1]:

    . . (2a + 3b)^3 \;=\;(2a)^3 +3(2a)^2(3b) + 3(2a)(3b)^2 + (3b)^3

    n . . . . . . . . =\; 8a^3 + 3(4a^2)(3b) + 3(2a)(9b^2) + 27b^3

    n . . . . . . . . =\;8a^3 + 36a^2b + 54ab^2 + 27b^3
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  4. #4
    Senior Member Paze's Avatar
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    Re: Binomial Theorem with coefficients

    Quote Originally Posted by veileen View Post
    Well, there's no difference.

    (a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k

    (2a+3b)^n = \sum_{k=0}^{n}\binom{n}{k}(2a)^{n-k}(3b)^k=\sum_{k=0}^{n}\binom{n}{k}2^{n-k}a^{n-k}\cdot 3^kb^k

    Um, did this help?
    Hello, Paze!

    I can get the binomial theorem to work just fine when doing
    but it becomes a problem as soon as I put coefficients, e.g.:

    Why? What is the difference and how do I calculate with coefficients?

    Consider .[1]

    Suppose we are given: .

    Then we have: .


    Substitute into [1]:

    . .

    n . . . . . . . .

    n . . . . . . . .

    It did indeed help. I was being silly and kept writing a^(n-k)*b^k even though I had coefficients.

    Thanks a lot!

    Where do I find a guide to write math like you guys do?
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  5. #5
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    Re: Binomial Theorem with coefficients

    Quote Originally Posted by Paze View Post
    Where do I find a guide to write math like you guys do?


    You can use LaTeX tags. Go to this tutorial.

    [tex] x = \neg A\cdot \neg B\cdot C + \neg A\cdot B\cdot \neg C + \neg A\cdot B\cdot C + A\cdot \neg B\cdot \neg C [/tex] gives  x = \neg A\cdot \neg B\cdot C + \neg A\cdot B\cdot \neg C + \neg A\cdot B\cdot C + A\cdot \neg B\cdot \neg C
    Click on the “go advanced” tab. On the toolbar you will see \boxed{\Sigma} clicking on that give the LaTeX wraps, [tex] [/tex]. The code goes between them.

    You can also click on Reply with Quote. That will allow you to ‘steal’ any code that you see that you like.
    Last edited by Plato; January 12th 2013 at 03:32 PM.
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