A Binomial is an algebraic expression of two terms which are connected by the operation: + or −.
For example: x+2y, x−y, x3+4y, a+b
Finding squares and cubes of a binomial by actual multiplication is not difficult. But the process of finding the expansion of binomials with higher powers becomes more difficult. There is a general formula which can help in finding the expansion of binomials with higher powers.
Binomial Theorem for Positive Integral Index
Tr+1 = nCr xn−r ar
- There are (n+1) terms in the expansion of (x+a)n.
- The degree of x in each term decreases while that of a increases such that the sum of the powers in each term is equal to n.
- nC0, nC1, nC2, nCr, nCn are called binomial coefficients.
- Since nCr = nCn−r, the coefficients of terms equidistant from the beginning and the end are equal.
- The binomial coefficients of the various terms of the expansion of (x+a)n for n = 1, 2, 3 form a pattern.
The arrangement of the binomial coefficients is known as Pascal’s triangle. The first and last numbers are 1 each. The other numbers are obtained by adding the left and right numbers in the previous row.