Discuss The Application of Binomial Theorem
An equation of the type y = f(x) is said to algebraic if it expressed in the form of fnyn +fn…1+……+f1y1+f0 = 0
Where f1 is an ith order polynomial in x.
The general form of equation f(x,y) = 0.Polynomial Equation:
Polynomial equations are simple class of algebraic equations that are represented as follows:
anxn + an-1xn-1 +……….+a1x+a0 = 0
The equation is called nth degree polynomial and has n roots.
Roots of the equation may be:
- Real and Different
- Real and Repeated
- Complex Numbers
An Algebraic Expression consisting of two terms which are related with ‘plus’ or ‘minus’ sign is called binomial expression.
2a + 3b, x2
Binomial theorems are given to any index form. The theorem which expands binomial term to any power is called binomial theorem. Lists of binomial theorem are given below:
Properties of Binomial Theorem:
- nCr =
- (a + b)n = nCn an + nCn-1 an-1 b + nCn-2 an-2 b2+…………..+ nC1 abn-1+ nC0 bn
- (a - b)n = nCn an - nCn-1 an-1 b - nCn-2 an-2 b2 -…………..+(-1)n-1 nC1 abn-1+(-1)n nC0 bn
- The number of terms in the expansion is one more than the index n.
- Sum of indices of term x and a is n.
- The coefficient of terms are nC0, nC1, nC2,…… nCn
- The coefficients of terms are of equivalent value.
Binomial coefficient are evaluated using Pascal’s triangle, the coefficient nC0, nC1, nC2,…… nCn are called binomial coefficient.
The number of terms of an expansion (x + a)n
is (n + 1).
Application of Binomial Theorem:
- To simply and to expand algebraic expression
- Numerical estimation.
- To solve the expansion of the term