BINOMIAL EXPANSION MADE EASY- Research and formulation by: Uday Kumar

INTRODUCTION:

In earlier classes, students learn how to find the squares and cubes of binomials like a+b and a-b. using them we can evaluate the numerical values of numbers like (98)2 = (100-2)2, (999)3= (1000-1)3 etc.

However, for higher powers like (95)5, (101)6, etc the calculations become difficult due to repeated multiplication. This difficulty was overcome by a theorem known as the binomial theorem.

In the traditional way given by binomial theorem, methods of permutation and combination n_(C_r ) which is only known to class 11+ students.

So here we have come up with a much easier way that requires no knowledge of permutation and combination and can easily applied to any person in class 5 to PHD level!

That is defined by:

(a+b)^n=a^n b^0+(na^((n-1)) b^1)/1+(n(n-1) a^((n-2)) b^2)/2+⋯a^0 b^n

Let’s assume the question (a+b)^2=?

Step 1: assign the highest power to ‘a’ and the lowest power to ‘b’

(a+b)^2= a^2 b^0

= a^2 (anything raised to zero is 1)

Step 2: for the second term, the power of ‘a’ becomes the coefficient that is constant for all the further terms and its power will reduce by 1 i.e. na(n-1) and for ‘b’ the power will be increased by 1 i.e. b(n+1)

Simultaneously, find the total number of preceding terms and divide by that.

Step 3: This will continue till the power of ‘a’ becomes zero and the power of ‘b’ becomes maximum.

(a+b)^2=a^2 b^0+(2a^1 b^1)/1+(2×a^0 b^2)/2

=a^2+2ab + b2

Similarly, we can solve for ‘n’ to be any number between 1 to infinity

Considering another example,

(a+b)^3=a^3 b^0+(3a^2 b^1)/1+(3×2a^1 b^2)/2+(3×1a^0 b^3)/3

=a^3+3a^2 b + 3ab^2+ b^3

Considering another example,

(a+b)^4=a^4 b^0+(4a^3 b^1)/1+(4×3×a^2 b^2)/2+(4×3×a^1 b^3 )/3+(4a^0 b^4 )/4

=a^4+4a^3 b^1+6a^2 b^2 + 4ab^3+ b^4

Similar ways can be followed for solving a binomial with negative signs also by replacing the negative term with its sign everywhere.

(a-b)^n=〖[a+(-b)]〗^n=a^n+(-b^0)+(na^(n-1) 〖(-b)〗^1⋅)/1+(n(n-1) a^(n-2) 〖(-b)〗^2)/2+⋯a^0 (-〖b)〗^n

Hence, we say, MATHS IS ALL ABOUT PATTERN!