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Homework Help: Binomial Theorem

  1. Dec 26, 2004 #1
    Binomial Theorem......

    Hi

    I need to know about binomial theorem....

    eg. how to in general expand (a+b)^n

    I dont understand the combinations / permutations....?


    thanks

    Roger
     
  2. jcsd
  3. Dec 26, 2004 #2
  4. Dec 26, 2004 #3
    And this thread.
     
  5. Dec 26, 2004 #4
    Another expansion can be obtained using the Taylor Function.

    [tex]f_n(x)=f(0)+f'(0)x+\frac{f''(0)x^2}{2!}+\frac{f^(3)(0)x^3}{3!}+...+\frac{f^(n)(0)x^n}{n!}+...[/tex]

    So in your case,

    [tex]f(x) = (a+b)^x[/tex]

    P.S. If I remember correctly, this only applies for real numbers.
     
  6. Dec 27, 2004 #5
    what i dont understand is the combinations and permutations........
    when used in binomial theorem ?

    what does it mean n choose x ?

    roger
     
  7. Dec 27, 2004 #6
    the thing I am used to is

    [tex] ^n C _r [/tex]

    i just use the one in my calculator, yours should have too. in fact, my teacher taught us the binomial theorem before permuations and combinations !
     
  8. Dec 27, 2004 #7
    Lets just expand some simple brackets first:
    [tex](a+b)^0 = 1[/tex]
    [tex](a+b)^1 = a+b[/tex]
    [tex](a+b)^2 = (a+b)(a+b) = a^2+ab+ba+b^2 = a^2+2ab+b^2[/tex]
    [tex](a+b)^3 = (a+b)(a+b)(a+b) = (a+b)(a^2+2ab+b^2) =

    a^3+2a^2b+ab^2+a^2b+2ab^2+b^3 = a^3+3a^2b+3ab^2+b^3[/tex]
    As you can see I have expanded the brackets the normal (well normal to me) way to do so. The problem that the binomial solves is when you have [tex](a+b)^1^5[/tex]or something in that nature.

    Also it is possible to see that there is a pattern to the coefficents (the numbers before the [tex]ab[/tex] or the [tex]ab^2[/tex]). The pattern, if wirtten out, is a pascal triangle. The top row has a single 1 then row two as two 1's (either side of the original one) and so on. This like will show you what I mean and then look back at the brackets above. The coefficents match (http://mathworld.wolfram.com/PascalsTriangle.html).

    Lets take the [tex](a+b)^2[/tex] and work it out using the binomial theorem.
    The first part is [tex]a^2[/tex]. We know this without any working out that occurs on paper but using the binomial it would be:
    [tex](a)^2[/tex] which gives [tex]a^2[/tex].
    The [tex]2ab[/tex] part is denoted by [tex]2(a^1b^1)[/tex] which gives [tex]2ab[/tex].
    The last part of this is similar to the first but using b instead of a.

    The pattern is: [tex](a+b)^2 = ^2 C _0(a^2(b)^0) + ^2 C _1(a^1(b)^1) + ^2 C _2(a^0(b)^2) = 1(a^2(1)) + 2(a(b)) + 1(1(b^2)) = a^2+2ab+b^2[/tex]
    As you can see, the indices add up to the original indice.

    A harder one just to show what the binomial does:
    [tex](a+b)^6 = ^6 C _0(a^6(b)^0) + ^6 C _1(a^5(b)^1) + ^6 C _2(a^4(b)^2) + ^6 C _3(a^3(b)^3) + ^6 C _4(a^2(b)^4) + ^6 C _5(a^1(b)^5) + ^6 C _6(a^0(b)^6)[/tex]
    [tex]= 1(a^6(1)) + 6(a^5(b)) + 15(a^4(b)^2) + 20(a^3(b)^3) + 15(a^2(b)^4) + 6(a^1(b)^5) + 1(a^0(b)^6)[/tex]
    [tex]= a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6[/tex]

    To finish the reason that I put the [tex]b[/tex] term in brackets without the indice is because if you get [tex](a-b)^n[/tex] then the part of the equation could be negative (e.g. [tex]20(a^3(-b)^3) = -20a^3b^3[/tex]).

    I hope this helps.

    The Bob (2004 ©)
     
    Last edited: Dec 28, 2004
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