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Introduction

The Binomial Theorem describes the algebraic expansion of the powers of a binomial (a polynomial of two terms), like $(x+y)^n$. The Binomial Theorem, was known to Indian and Greek mathematicians in the 3rd century B.C. for some cases. The credit for the result for natural exponents goes to the Arab poet and mathematician Omar Khayyam (A.D. 1048-1122). Further generalisation to rational exponents was done by the British mathematician Newton (A.D. 1642-1727).

Let us write out by hand a number of cases to see if we can find a pattern: \begin{align*} (x+y)^1 &= x + y \\ (x+y)^2 &= x^2 + 2xy + y^2 \\ (x+y)^3 &= x^3 + 3x^2y+ 3xy^2 + y^3 \\ (x+y)^4 &= x^4 + 4x^3y + 6x^2y^2+4xy^3+y^4 \\ (x+y)^5 &= x^5 + 5x^4y + 10x^3y^2+10x^2y^3+5xy^4+ y^5 \end{align*}

Before we continue with the Binomial Theorem take a refresh of some combinatorial definitions.

Bionomial Theorem Positive Exponents

Now we are in the position to state the bionomial theorem for the case of $1$ and $x$.

Proposition

Let n be a positive integer: \[ (1+x)^n=\sum_{r=0}^n \binom{n}{r}x^r \]

Proof

Proposition

Let n be a positive integer: \[ (1-x)^n=\sum_{r=0}^n \binom{n}{r}(-1)^rx^r \]

Proof

More generally, we can state the theorem for any $x$ and $y$

Proposition

Let n be a positive integer: \[ (x+y)^n=\sum_{r=0}^n \binom{n}{r}x^ry^{n-r} \]

Proof

Binomial Theorem for Negative Exponents

Next we want to generalize the theorem for negative exponents.

Proposition

Let n be a positive integer: \[ (1-x)^{-n}=\sum_{r=0}^\infty \binom{n+r-1}{r}x^r \]

Proof

Again this result can be generalized.

Proposition

Let n be a positive integer: \[ (x+y)^{-n}=\sum_{r=0}^\infty \binom{n+r-1}{r}(-1)^rx^ry^{n-r} \]

Proof

Multinomial Theorem

We generalize the Binomial Theorem to the algebraic expansion of a multinomial polynomial.

Proposition

Let $n$ be a positive integer: \[(x_1+x_2+\cdots+x_k)^n=\sum_{\substack{n_1,n_2,\cdots,n_k>0 \\ n_1+n_2+\cdots+n_k=n}}{\binom{n}{n_1,n_2,\dots,n_k}x_1^{n_1}x_2^{n_2}\cdots x_k^{n_k}}\]

Proof

Newton's Bionomial Theorem

Netwon generalized the binomial theorem $(x+y)^\alpha$ for any real number $\alpha$.

Proposition

Let $\alpha \in \mathbb{R}$, $0\leq \abs{x} \less \abs{y}$ then \[ (x+y)^\alpha=\sum_{k=0}^\infty{\binom{\alpha}{k}x^ky^{\alpha-k}} \] with \[ \binom{\alpha}{k}=\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!} \]

Proof