Binomial Expansions

The sum $a+b$ is called a binomial as it contains two terms.
Any expression of the form $(a+b)^n$ is called a power of a binomial.
All binomials raised to power can be expanded using the same general principles. In this lesson, therefore, we consider the expansion of the general expression $(a+b)^n$ where $n \in \mathbb{N}$.
Consider the following algebraic expressions.
\( \begin{align} \displaystyle
(a+b)^1 &= a^1 + b^1 \\
&= a+b \\
(a+b)^2 &= a^2 + 2a^1b^1 + b^2 \\
&= a^2 + 2ab + b^2 \\
(a+b)^3 &= a^3 + 3a^2b^1 + 3a^1b^2 + b^3 \\
&= a^3 + 3a^2b + 3ab^2 + b^3 \\
\end{align} \)
In the theory of Pascal’s Triangle, the coefficients of binomial expansions come from Pascal’s Triangle.
\begin{matrix}
n=0&&&&&&1 \\
n=1&&&&&1&&1 \\
n=2&&&&1&&2&&1 \\
n=3&&&1&&3&&3&&1 \\
n=4&&1&&4&&6&&4&&1 \\
\end{matrix}
Let’s consider the expansion of $(a+b)^4$.
This expansion produces $5$ terms; $a^4;a^3b^1;a^2b^2;a^1b^3;b^4$.
Their corresponding coefficients are $1;4;6;4;1$, respectively as per Pascal’s Triangle.
Therefore we can obtain the expansion of $(a+b)^4$ by combining these terms and coefficients.
\( \begin{align} \displaystyle
(a+b)^4 &= 1a^4 + 4a^3b^1 + 6a^2b^2 + 4a^1b^3 + 1b^4 \\
&= a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \\
\end{align} \)
Example 1
Expand $(a+b)^5$.
\( \begin{align} \displaystyle
\begin{matrix}
n=0&&&&&&1 \\
n=1&&&&&1&&1 \\
n=2&&&&1&&2&&1 \\
n=3&&&1&&3&&3&&1 \\
n=4&&1&&4&&6&&4&&1 \\
n=5&1&&5&&10&&10&&5&&1 \\
\end{matrix}
\end{align} \)
\( \begin{align} \displaystyle
(a+b)^5 &= 1a^5 + 5a^4b^1 + 10a^3b^2 + 10a^2b^3 + 5a^1b^4 + 1b^5 \\
&= a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
\end{align} \)
Example 2
Expand $(a-b)^5$.
\( \begin{align} \displaystyle
\begin{matrix}
n=0&&&&&&1 \\
n=1&&&&&1&&1 \\
n=2&&&&1&&2&&1 \\
n=3&&&1&&3&&3&&1 \\
n=4&&1&&4&&6&&4&&1 \\
n=5&1&&5&&10&&10&&5&&1 \\
\end{matrix}
\end{align} \)
\( \begin{align} \displaystyle
(a-b)^5 &= 1a^5 + 5a^4(-b)^1 + 10a^3(-b)^2 + 10a^2(-b)^3 + 5a^1(-b)^4 + 1(-b)^5 \\
&= a^5-5a^4b + 10a^3b^2-10a^2b^3 + 5ab^4-b^5
\end{align} \)
Example 3
Expand $(a+2b)^5$.
\( \begin{align} \displaystyle
\begin{matrix}
n=0&&&&&&1 \\
n=1&&&&&1&&1 \\
n=2&&&&1&&2&&1 \\
n=3&&&1&&3&&3&&1 \\
n=4&&1&&4&&6&&4&&1 \\
n=5&1&&5&&10&&10&&5&&1 \\
\end{matrix}
\end{align} \)
\( \begin{align} \displaystyle
(a+2b)^5 &= 1a^5 + 5a^4(2b)^1 + 10a^3(2b)^2 + 10a^2(2b)^3 + 5a^1(2b)^4 + 1(2b)^5 \\
&= a^5 + 5a^4 \times 2b + 10a^3 \times 4b^2 + 10a^2 \times 8b^3 + 5a \times 16b^4 + 32b^5 \\
&= a^5 + 10a^4b + 40a^3b^2 + 80a^2b^3 + 80ab^4 + 32b^5
\end{align} \)
Example 4
Expand $\bigg(a+\dfrac{1}{a}\bigg)^3$.
\( \begin{align} \displaystyle
\begin{matrix}
n=0&&&&&1 \\
n=1&&&&1&&1 \\
n=2&&&1&&2&&1 \\
n=3&&1&&3&&3&&1 \\
\end{matrix}
\end{align} \)
\( \begin{align} \displaystyle
\bigg(a+\dfrac{1}{a}\bigg)^3 &= 1a^3 + 3a^2 \times \dfrac{1}{a^1} + 3a^1 \times \dfrac{1}{a^2} + \dfrac{1}{a^3} \\
&= a^3 + 3a + \dfrac{3}{a} + \dfrac{1}{a^3}
\end{align} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses