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

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

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Divisibility Proof Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

 



Your email address will not be published. Required fields are marked *