# Binomial Expansion | Binomial Theorem

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.

## 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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