Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is binomial theorem

  1. Jul 23, 2014 #1
    Definition/Summary

    The binomial theorem gives the expansion of a binomial [itex](x+y)^n[/itex] as a summation of terms. The binomial theorem for positive integral values of 'n', is closely related to Pascal's triangle.

    Equations

    The theorem states, for any [itex]n \; \epsilon \; \mathbb{N}[/itex]

    [tex](x+y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 +......+\binom{n}{n}x^0y^n[/tex]

    In summation form,
    [tex](x+y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r}y^r[/tex]


    Cases

    1. Substituting y=-y we get,

    [tex](x-y)^n = \binom{n}{0}x^ny^0 - \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 -......+ (-1)^{n}\binom{n}{n}x^0y^n[/tex]

    2. Having y=1 gives,


    [tex](x+1)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1} + \binom{n}{2}x^{n-2} +......+ \binom{n}{n}x^0[/tex]

    Extended explanation

    Proof by Induction

    When [itex]n=0[/itex], the statement obviously holds true, giving [itex](x+y)^0= \binom{0}{0}=1[/itex]

    Assuming it to be true for [itex]n=k[/itex]

    [tex](x+y)^n = \sum_{r=0}^{k} \binom{n}{k} x^{n-k}y^k[/tex]

    Now it needs to hold for [itex]n=k+1[/itex] to complete the inductive step. We use

    [tex](x+y)^{k+1} = x(x+y)^k + y(x+y)^k[/tex]

    Expanding each [itex](x+y)^k[/itex] individually, multiplying by x and y respectively,

    [tex](x+y)^{k+1} = \sum_{r=0}^{k} x^{k+r-1}y^r + \sum_{r=0}^{k} x^{k+r}y^{r+1}[/tex]

    Using the property,

    [tex]\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}[/tex]

    We get,
    [tex](x+y)^{k+1} = \sum_{r=0}^{k+1} \binom{k}{r} x^{(k+1)-r}y^r[/tex]

    This completes our inductive step, proving the theorem.


    Generalization

    For any value of 'n', whether positive, negative, or fractional, the binomial expansion is given by,

    [tex](x+y)^n = x^n + nx^{n-1}y+ \frac{n(n-1)}{2}x^{n-2}b^2 + ......+b^n[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: What is binomial theorem
  1. Binomial Theorem (Replies: 1)

  2. The Binomial Theorem (Replies: 2)

  3. Binomial Theorem (Replies: 6)

  4. Binomial Theorem (Replies: 3)

  5. Binomial theorem (Replies: 3)

Loading...