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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 f_{n}y_{n} +f_{n…1}+……+f_{1}y_{1}+f_{0} = 0
Where f_{1} is an i^{th }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:
Polynomial Equation:
Polynomial equations are simple class of algebraic equations that are represented as follows:
a_{n}x^{n} + a_{n1}x^{n1 }+……….+a_{1}x+a_{0} = 0
The equation is called nth degree polynomial and has n roots.
Roots of the equation may be:
Example: 2a + 3b, x^{2} + 4y
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, x^{2} + 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:
Properties of Binomial Theorem:
 ^{n}C_{r} =
 (a + b)^{n} = ^{n}C_{n} a^{n} + ^{n}C_{n1} a^{n1 }b + ^{n}C_{n2} a^{n2} b^{2}+…………..+ ^{n}C_{1} ab^{n1}+ ^{n}C_{0} b^{n}
 (a  b)^{n} = ^{n}C_{n} a^{n}  ^{n}C_{n1} a^{n1} b  ^{n}C_{n2} a^{n2} b^{2} …………..+(1)^{n1} ^{n}C_{1} ab^{n1}+(1)^{n} ^{n}C_{0} b^{n}
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 ^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2},…… ^{n}C_{n}
 The coefficients of terms are of equivalent value.
Binomial Coefficient:
Binomial coefficient are evaluated using Pascal’s triangle, the coefficient ^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2},…… ^{n}C_{n} 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