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
General Term In Binomial Expansion:
The binomial expansion of (x + y)n,
(x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr or (1 + x)n is nCr xr
Some Binomial Expansions:
- (x + y)n + (x−y)n = 2[C0 xn + C2 xn-1 y2 + C4 xn-4 y4 + …]
- (x + y)n – (x−y)n = 2[C1 xn-1 y + C3 xn-3 y3 + C5 xn-5 y5 + …]
- (1 + x)n = nΣr-0 nCr . xr = [C0 + C1 x + C2 x2 + … Cn xn]
- (1+x)n + (1 − x)n = 2[C0 + C2 x2+C4 x4 + …]
- (1+x)n − (1−x)n = 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. nC0 = nCn, nC1 = nCn − 1, nC2 = nCn – 2 .
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
- (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.
Q1. What is the value of (2+5)3?
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
= 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