The binomial expansion is a fundamental concept in Maths, offering a systematic approach to expanding expressions raised to a power. This method is particularly valuable for simplifying expressions with high exponents, where direct multiplication is not feasible.

Understanding the Binomial Theorem for Positive Integer Exponents

The binomial theorem facilitates the expansion of expressions in the form $(a+b)^n$, where $n$ is a positive integer. The expansion involves a sum of terms featuring binomial coefficients.

Binomial Coefficients and Factorial Notation

• Binomial coefficients are represented as $\left(\begin{array}{c}n \ r\end{array}\right)$, where $n$ is the exponent and $r$ ranges from 0 to $n$.
• These coefficients, also written as $^nC_r$, are calculated using the formula $\frac{n!}{r!(n-r)!}$, where $n!$ denotes the factorial of $n$.

Expansion Formula

The binomial expansion of $(a+b)^n$ is: $(a+b)^n = \sum_{k=0}^n \left(\begin{array}{c}n \ k\end{array}\right) a^{n-k} b^k$ This summation notation provides an alternative way to express the expanded form.

Key Patterns in Binomial Expansion

1. Symmetry: $^nCr = ^nC{n-r}$.
2. Boundary Values: $^nC_0 = ^nC_n = 1$ and $^nC1 = ^nC{n-1} = n$.
3. Pascal's Triangle is a useful tool for visualising these patterns, with$n$indicating the row (starting from 0) and $r$ the position from the left (also starting from 0).

Worked Examples

Example 1: Basic Expansion

Expand the expression $(a+b)^5$.

Solution:

Applying the binomial theorem:

Example 2: Expansion with Coefficients

Expand the expression $(3x+2)^4$.

Solution:

Using the binomial theorem:

Example 3: Finding Specific Coefficients

Find the coefficient of $x^3$ in the expansion of $(1+2x)^8$.

Solution:

Identifying the term with $x^3$ in the binomial expansion:

Therefore, the coefficient of $x^3$ is 448.