Difference between revisions of "Binomial Theorem"
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==Usage== | ==Usage== | ||
| − | Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial <math>x^5+4x^4+6x^3+4x^2+x</math>, | + | Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial <math>x^5+4x^4+6x^3+4x^2+x</math>, one could factor it as such: <math> x(x^4+4x^3+6x^2+4x+1)=x(x+1)^{4}</math>. It is a good idea to be familiar with binomial expansions, and knowing the first few coefficients would also be beneficial. |
==See also== | ==See also== | ||
*[[Combinatorics]] | *[[Combinatorics]] | ||
Revision as of 13:47, 3 July 2006
The Theorem
First discovered by Newton, the Binomial Theorem states that for real or complex a,b,
.
This may be shown for the integers easily:
.
Repeatedly using the distributive property, we see that for a term 
, we must choose 
of the 
terms to contribute an 
to the term, and then each of the other 
terms of the product must contribute a 
. Thus, the coefficient of is 
. Extending this to all possible values of from 
to , we see that .
Usage
Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial 
, one could factor it as such: 
. It is a good idea to be familiar with binomial expansions, and knowing the first few coefficients would also be beneficial.
