Binomial Theorem

An algebraic expression that contains two terms is known as a binomial expression. Its general form is (x + y), whereas the binomial theorem involves expanding the expression to 'n'. In other words, the expansion of (x + y)n is called the binomial theorem.

Binomial Theorem Definition

Simply put, the binomial theorem is actually a formula used for finding any power of a binomial without having to multiply it at length. The theorem actually provides a way to calculate the product (x + y)n for any positive integer using formula:   

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

where, $\binom{n}{r}$ = $\frac{n!}{r!(n - r)!}$ is binomial coefficient.

Let us substitute 1 for x and x instead of y, then we get

$(1+x)^n=\begin{pmatrix} n\\ 0 \end{pmatrix}$ x0 + $\begin{pmatrix} n\\ 1 \end{pmatrix}$ x1 + $\begin{pmatrix} n\\ 2 \end{pmatrix}$ x2 + $\begin{pmatrix} n\\ 3 \end{pmatrix}$ x3 + …+ $\begin{pmatrix} n\\ n - 1 \end{pmatrix}$xn-1 + $\begin{pmatrix} n\\ n \end{pmatrix}$xn

It is given by

(1+x)n = $\sum_{k=0}^{n}$ $\begin{pmatrix} n\\ k \end{pmatrix}$ xk  

The Quick way to expand the binomials using the binomial theorem is given by   

$\begin{pmatrix} n\\ k + 1 \end{pmatrix}$ = ($\frac{n-k}{k+1}$)$\begin{pmatrix} n\\ k \end{pmatrix}$

Binomial Theorem History

It was back in 300BC when the binomial theorem became known for the case n = 2 by Euclid. Pascal again described it in modern form in 1665, whereas he came up with the triangular arrangement of the binomial coefficients during 17th century. Newton also played his part and presented a similar formula for negative integers - his shortcut really helped simplify the process of multiplying binomials, especially when used with specific numbers explained in Pascal's triangle.

Binomial Theorem Formula 

Binomial theorem can expand any power of x + y into a sum of the form given by the following formula:

The formula can be written using the notion of summation as

(x + y)n = $\sum_{k=0}^{n}$ $\begin{pmatrix} n\\ k \end{pmatrix}$ xn - kyk

Here, $\binom{n}{r}$ denotes the respective coefficient. The coefficients in the binomial expansion are called as the binomial coefficients.

Binomial expression can also be expanded as:

(x + y)n = nC0 xn y0 + nC1 xn - 1y1 + nC2 xn - 2y2 + ......... + nCrxn - ryr + ......... + nCn - 1x1yn - 1 + nCnx0yn .

The Algebra Formula for Binomial Theorem is,

$(a+b)^n=a^nb^0+na^{n-1}b^1+\frac{n(n-1)}{2!}a^{n-2}b^2+...+a^0b^n$

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