Content Binomial Theorem

## 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*}

• The right hand side is called the binomial expansion of the left hand side.
• We have written the power of a binomial in the expanded form in such a way that the terms are in descending powers of the first term of the binomial
• The number of terms in the expansion is one more than the exponent of the binomial.
• The exponent of $x$ in the first term is the same as the exponent of the binomial, and the exponent decreases by 1 in each successive term of the expansion.
• The exponent of $y$ in the first term is zero (as $y^0= 1$). The exponent of $y$ in the second term is 1, and it increases by 1 in each successive term till it becomes the exponent of the binomial in the last term of the expansion
• The sum of the exponents of x and y in each term is equal to the exponent of the binomial.

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: $(1+x)^n=\sum_{r=0}^n \binom{n}{r}x^r$

### 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: $(x+y)^n=\sum_{r=0}^n \binom{n}{r}x^ry^{n-r}$

## 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: $(x+y)^{-n}=\sum_{r=0}^\infty \binom{n+r-1}{r}(-1)^rx^ry^{n-r}$

### Multinomial Theorem

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

### Proposition

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

### 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 $\binom{\alpha}{k}=\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}$