# Binomial Theorem Formula

## Binomial Theorem Formula

The expanded value of an algebraic expression of the form (x + y)n is calculated using the Binomial Theorem Formula. It is simple to get the values of (x + y)2, (x + y)3, and (a + b + c)2 by algebraically multiplying the numbers according to the exponent value. The expanded version of (x + y)17 or other similar expressions with larger exponential values, on the other hand, requires far too much computation. The Binomial Theorem Formula can assist in making things simpler.

This Binomial Theorem Formula expansion’s exponent value can be a fraction or a negative number. Here, students only need to consider explanations for positive values. On the Extramarks website and mobile application, with the study materials on Binomial Theorem Formula, students can study more about the Binomial Theorem Formula, formula, and qualities of coefficients.

## What is Binomial Expansion?

In the fourth century BC, a well-known Greek mathematician by the name of Euclid made the first reference to the Binomial Theorem Formula. The binomial theorem can be used to expand the algebraic expression (x + y)n, which represents it as a sum of terms with separate exponents for the variables x and y.Each word in the Binomial Theorem Formula has a coefficient, which is a numerical value.

### Binomial Theorem Formula

The algebraic terms of the form (x + y)n are expanded using the Binomial Theorem Formula. (x+y)n = nC0 xny0 + nC1 xn-1y1,… + nCn-1 x1yn-1 + nCn x0yn. In this case, there are n + 1 terms in the binomial expansion with an exponent of n. In a progressive way, the second term’s exponent is gradually rising while the first term’s exponent in the expansion is steadily declining. Use Pascal’s triangle or the combination formula nCr = n! / [r! (n – r)!]! to find the coefficients of the Binomial Theorem Formula.

### Properties of the Binomial Expansion

Here are the properties of the Binomial Theorem Formula:

• The Binomial Theorem Formula of (x+y)n contains (n+1) terms in total.
• The exponents of x and y added together are always n.
• Binomial coefficients are also known as C0, C1, C2,…, and Cn and are also represented by nC0, nC1, nC2,…, and nCn.
• The binomial coefficients that are equally spaced from the beginning and the end are equal; for example, nC0 = nCn, nC1 = nCn-1, nC2 = nCn-2, etc.

### Binomial Series

Students can find detailed notes based on the Binomial Theorem Formula on the Extramarks website and mobile application. Students can look up the study materials before their final examinations to be prepared and to score well. The study materials are very comprehensively designed, keeping the complex needs of students in mind. Therefore, students can trust the accuracy of study materials.

## Solved Examples

1. Determine x if the binomial expansion’s third term equals 2560.

Solution:  ⇒ (log2x)2 = 4

0 or -2 for log2x

⇒ x = 4 or 1/4.