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Binomial Expansion Formula

In algebra, the algebraic expansion of powers of a binomial is expressed by binomial expansion. In binomial expansion, a polynomial (x + y)n is expanded into a sum involving terms of the form a x + b y + c, where b and c are non-negative integers, and the coefficient a  is a positive integer depending on the value of n and b. Let us learn more about the binomial expansion formula.

Binomial Expansion Formula

Terms of Binomial Expansion

Binomial expansion specifies the expansion of a binomial. It is also known as binomial theorem. In binomial expansion, we find the middle term. The different terms used in the binomial expansion are

  • General Term
  • Middle Term
  • Independent Term
  • Determining a Particular Term
  • Numerically greatest term
  • Ratio of Consecutive Terms/Coefficients

Binomial Expansion Formula

General Term In Binomial Expansion:

The binomial expansion of (x + y)n,

(x + y)= nC0 x+ nC1 xn-1 . y + nC2 xn-2 . y+ … + nCn yn

General Term = Tr+1 = nCr xn-r . yr or (1 + x)n is nCxr

Some Binomial Expansions:

  • (x + y)+ (x−y)= 2[C0 x+ C2 xn-1 y+ C4 xn-4 y+ …]
  • (x + y)– (x−y)= 2[C1 xn-1 y + C3 xn-3 y+ C5 xn-5 y+ …]
  • (1 + x)n  nΣr-0 nC. x= [C+ C1 x + C2 x+ … Cn xn]
  • (1+x)+ (1 − x)= 2[C0 + C2 x2+C4 x+ …]
  • (1+x)− (1−x)= 2[C1 x + C3 x3 + C5 x5 + …]

Properties of Binomial Expansion

Properties of binomial expansion are:

  • In binomail expansion of (x+y)n number terms are (n+1)
  • The sum of exponents of x and y is always n.
  • nC0, nC1, nC2, … nCn are called binomial coefficients and also represented by C0, C1, C2, … Cn
  • The binomial coefficients in the beginning and in the end are equal i.e. nC= nCn, nC= nC− 1, nC= nC– 2 .

Pascal’s Triangle

A triangular array of the binomial coefficients of the expression is known as Pascal’s Triangle. Pascal’s triangle contains the values of the binomial coefficient of the expression.

The expansion of (x + y) 2 is

(x + y) 2 = x 2 + 2xy + y 2

Hence,

  • (x + y) 3 = (x + y)(x + y) 2
  • = (x + y)(x 2 + 2xy + y 2 )
  • = x 3 + (1 + 2)x2y + (2 + 1)xy2 + y3
  • x 3 + 3x 2y + 3xy2 + y 3

In general we see that the coefficients of (x + y)n come from the n-th row of Pascal’s Triangle, in which each term is the sum of the two terms just above it.

Solved Example

Q1. What is the value of (2+5)3?
Solution:

The binomial expansion formula is,

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

From the given equation,

x = 2 ; y = 5 ; n = 3

(2+5)3

= 23 + 3(22)(51) + 3×22!(21)(52) + 3×2×13!(20)(53)

= 8 + 3(4)(5) + 62(2)(25) + 66(125)

= 8 + 60 + 150 + 125 = 343

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