# Binomial Expansion Formula

## Binomial Expansion Formula

A binomial statement can be multiplied or expanded quickly using the Binomial Expansion Formula. Significantly more expressiveness has been expressed with more intensity. It is known that multiplying such sentences with huge powers and phrases is challenging. However, in this context, Binomial Expansion Formula comes in quite handy. This article will examine the Binomial Expansion Formula. To understand the notion, students can first go over a few examples.

## What is Binomial Expansion, and How does It work?

The Binomial Expansion Formula is described by the Binomial Theorem, a mathematical statement. This theorem states that the polynomial (x+y)n can be expanded into a series of sums composed of terms of the type xbyc.

The requirement of the Binomial Expansion Formula is that b + c = n, and the exponents b and c are both non-negative integers. Additionally, each term’s coefficient is a different positive integer, depending on n and b.

Consider the following for n = 4:

(x+y)⁴=x⁴+4x³y+6x²y²+4xy³+y⁴

It goes without saying that manually multiplying such words and their expansions would be incredibly painful. Thankfully, a formula for this growth has been developed, which students can easily use.

### Binomial Expansion Formula of Natural Powers

This Binomial Expansion Formula provides the expansion of (x + y)n when n is a natural number.There are (n + 1) words in the expansion of (x + y)n.

### Binomial Expansion Formula of Rational Powers

When n is a rational number, the expansion of (1 + x)n is given by the Binomial Expansion Formula. There are a limitless number of terms in this extension.

(1 + x)

n = 1 + n x + [n(n – 1)/2!] x2 + [n(n – 1)(n – 2)/3!] x3 +...

### Solved Example

1. Apply the Binomial Expansion Formula to evaluate (3 + 7)3.

Solution: The following is the formula for binomial expansion:

(x+y)n = xn+nxn-1y+n(n1)2! xn-2y2+…..+ yn

In the mentioned issue,

x = 3 ; y = 7 ; n = 3

(3 + 7)3 = 33 + 3 x 32 x 7 + (3 x 2)/2! x 31 x 72 + 73

= 27 + 189 + 441 + 343

(3 + 7)3 = 1000

## 1. What is the Binomial Expansion Formula?

When a binomial is raised to a number, the Binomial Expansion Formula is utilised to determine the expansion. The Extramarks website and mobile application provide the Binomial Expansion Formula.

## 2. How can the Binomial Expansion Formula be derived?

The procedure to derive the Binomial Expansion Formula is as follows:

Step 1: Establish that n = 1 is true.

Step 2: Assume that n = k is a valid value.

Step 3 is to validate the n = k equation.