Usually we will expand powers just by expanding it using normal multiplication or using the following formulae.
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Fine. We can use these formulae when the power is either 2 or 3. But, Is there any formula for the expansion of (a+b)^4? We may say we can expand it. Yes. We can expand though it is a bit difficult.
But what if the powers are big numbers like 5,6,7,8….?
We have a solution for this called “Binomial theorem” which is used for the expansion of binomials with different powers (higher powers even).
Binomial theorem is defined as follows.
(a+b)^n = nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b^2 + … + nCn b^n
(a-b)^n = nC0 a^n – nC1 a^(n-1) b + nC2 a^(n-2) b^2 – … + (-1)^n nCn b^n
The process of expansions using Binomial theorem is explained through the following examples.
Example 1:
Expand (x+y)^5 using binomial theorem.
Solution:
Substituting a=x, b=y and n=5 in the expansion of binomial theorem,
(x+y) 5 = 5C0 x^5 + 5C1 x^4 y + 5C2 x^3 y^2 + 5C3 x^2 y^3 + 5C4 x y^4 + 5C5 y^5
= x^5 + 5x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5
(Here, 5C0=1, 5C1=5, 5C2=10, 5C3=10, 5C4=5, 5C5=1).
Example 2:
Expand (x-y)^5 using binomial theorem.
Solution:
Substituting a=x, b=y and n=5 in the expansion of binomial theorem,
(x-y) 5 = 5C0 x^5 – 5C1 x^4 y + 5C2 x^3 y^2 – 5C3 x^2 y^3 + 5C4 x y^4 – 5C5 y^5
= x^5 – 5x^4 y + 10 x^3 y^2 – 10 x^2 y^3 + 5 x y^4 – y^5
(Here, 5C0=1, 5C1=5, 5C2=10, 5C3=10, 5C4=5, 5C5=1).
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