Binomial Theorem

The Theorem

First invented by Newton, the Binomial Theorem states that for real or complex a,b,
$(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}$

This may be shown for the integers easily:
$\displaystyle (a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}$
Repeatedly using the distributive property, we see that for a term $\displaystyle a^m b^{n-m}$ , we must choose $m$ of the $n$ terms to contribute an $a$ to the term, and then each of the other $n-m$ terms of the product must contribute a $b$ . Thus the coefficient of $\displaystyle a^m b^{n-m}$ is $\displaystyle n \choose m$ . Extending this to all possible values of $m$ from $0$ to $n$ , we see that $(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}$ .

Usage

Many factorizations involve complicated polynomials with binomial coefficients. For example, If a contest problem involved the polynomial $x^5+4x^4+6x^3+4x^2+x$ , I would factor it as such: $x(x^4+4x^3+6x^2+4x+1)=x(x+1)^{4}$ . It is a good idea to be familiar with binomial expansions, and knowing the first few coefficients would also be beneficial.

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Binomial Theorem

The Theorem

Usage

See also