Binomial Expansion | Binomial Theorem

Binomial Expansion is based on two terms, that is binomial.
Any expression of the form $(a+b)^n$ is called the 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.

Question 1

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

\begin{aligned} \displaystyle (2x+3)^3 &= (2x)^3+3(2x)^23^1+3(2x)^13^2+3^3 \\ &= 8x^3 + 36x^2 +54x + 27 \\ \end{aligned}

Question 2

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

\begin{aligned} \displaystyle (x-4)^4 &= x^4 + 4x^3(-4)^1 + 6x^2(-4)^2 + 4x^1(-4)^3 + (-4)^4 \\ &= x^4 – 16x^3 + 96x^2 -256x + 256 \\ \end{aligned}

Question 3

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

\begin{aligned} \displaystyle \Big(x-\dfrac{2}{x}\Big)^5 &= x^5 + 5x^4\Big(\dfrac{-2}{x}\Big) + 10x^3 \Big(\dfrac{-2}{x}\Big)^2 + 10x^2\Big(\dfrac{-2}{x}\Big)^3 + 5x\Big(\dfrac{-2}{x}\Big)^4 + \Big(\dfrac{-2}{x}\Big)^5 \\ &= x^5 + 10x^3 + 40x – \dfrac{80}{x} + \dfrac{80}{x^3} – \dfrac{32}{x^5} \\ \end{aligned}

Question 4

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

\begin{aligned} \displaystyle \big(2-\sqrt{2}\big)^5 &= 2^5 – 5 \times 2^4\sqrt{2} + 10 \times 2^3\sqrt{2}^2 – 10 \times 2^2\sqrt{2}^3 + 5 \times 2 \sqrt{2}^4 – \sqrt{2}^5 \\ &= 32 – 80\sqrt{2} + 160 – 80\sqrt{2} + 40 – 4\sqrt{2} \\ &= 232 – 164\sqrt{2} \end{aligned}