Any polynomial with two terms in the expression is called a binomial. Binominal theorem is used to calculate the multiple powers (square, cube and so on) of that binomial expression.
In the formula above, we have addition of two terms (a and b). This binomial is raised to the power n.
Example: If n=2 then it is the square of (a+b) = (a+b) x (a+b)
Similarly, Power n of (a+b) = (a+b) x (a+b) x (a+b) x ….. n times.
Proving Binomial Theorem
>>> (a+b)x(a+b) = a^2 + 2ab + b^2
>>> (a+b)x(a+b)x(a+b) = (a^2 + 2ab + b^2)(a+b) = a^3 + 3a^2b + 3ab^2 + b^3
>>> (a+b)x(a+b)x(a+b)x(a+b) = (a^3 + 3a^2b + 3ab^2 + b^3)(a+b) = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
Here we have pattern, where power one term is decreasing and the power of second term is increasing. But the sum of both these power is always equal to the power of the binomial.
And we also know that zero power of any number is equal to 1. (a^0 = 1 ; b^0 = 1)
If we keep on increasing the power ( say n, in general) then the same pattern follows. We can write the formula for the pattern and the same formula is actually called the binomial theorem.
This formula consists of Sigma (summation) and Combination (nCk). Values of k varies from 0 to n.
∑ is called Sigma and it is used for showing the sum of various terms.
Now let us keep the values of n in the formula and see if it comes equal to our calculations.
for n = 1,
(a+b) = (1!/1!.0!). a^1 + (1!/0!.1!). b^1 = (1.a + 1.b) = (a+b)
for n =2,
(a+b)^2 = (2!/2!.0!. a^2. b^0 + 2!/1!.1!. a^1. b^1 + 2!/2!.0!. a^0. b^2) = (1.a^2 + 2.a.b + 1.b^2) = (a^2+2ab+b^2)
>>> (a+b)^2 = (a^2+2ab+b^2)
(Which is equal to the output of multiplications)
Similarly, we can find the higher powers of (a+b) with the help of binomial theorem.
If we have to find the higher powers of (a-b) then also we can use the binomial theorem.
We can rewrite (a-b) as (a+(-b)) and use the magical formula. We can easily calculate the higher powers without doing much calculations. This is power of this binomial theorem.