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 \( \big(2-\sqrt{2}\big)^5 \).

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

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