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Discuss The Application of Binomial Theorem

Algebraic Equation:
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
Binomial Expression:
An Algebraic Expression consisting of two terms which are related with ‘plus’ or ‘minus’ sign is called binomial expression.

Example: 2a + 3b, x2 + 4y

Binomial Theorem:
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:
  • 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

Properties of Binomial Theorem:
  • 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:
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