Binomial Expansion | Binomial Theorem

Binomial Expansion Series

Binomial Expansion is based on two terms, that is binomial.
Any expression of the form \( (a+b)^n \) is called power of a binomial.
All binomials raised to a power can be expanded using the same general principles.
\( \begin{aligned} \displaystyle
(a+b)^1 &= a+b \\
(a+b)^2 &= (a+b)(a+b) \\
&= a^2+2ab+b^2 \\
(a+b)^3 &= (a+b)(a^2+2ab+b^2) \\
&= a^3+3a^2b+3ab^2+b^3 \\
(a+b)^4 &= (a+b)(a^3+3a^2b+3ab^2+b^3) \\
&= a^4+4a^3b+6a^2b^2+4ab^3+b^4 \\
\end{aligned} \)

For the expansion of \((a+b)^2 \) where \( n \in \text{N}: \)

  • As we look from left to right across the expansion, the powers of a decrease by 1, while the powers of \(b\) increase by 1.
  • The sum of the powers of \(a\) and \(b\) in each term of the expansion is \(n\).
  • The number of terms in the expansion is \( n+1 \).
  • The coefficients of the terms are row \(n\) of Pascal’s triangle.


Binomial Expansion by Pascals Triangle

Practice Questions

Question 1

Find the binomial expansion of \( (2x+3)^3 \).

Question 2

Find the binomial expansion of \( (x-4)^4 \).

Question 3

Find the binomial expansion of \( \Big(x-\dfrac{2}{x}\Big)^5 \).

Question 4

Find the binomial expansion of \( \big(2-\sqrt{2}\big)^5 \).


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