Binomial Theorem

The Binomial Theorem provides a method to expand expressions of the form \((a+b)^n\), where \(a\) and \(b\) are any real numbers and \(n\) is a positive integer. The expansion is a sum of terms involving powers of \(a\) and \(b\) multiplied by binomial coefficients.

Formula Derivation

The binomial expansion of \((a+b)^n\) is given by:

\[ (a+b)^n = \sum_{r=0}^n {n \choose r} a^{n-r} b^r \]

where \({n \choose r}\) is the binomial coefficient defined as:

\[ {n \choose r} = \frac{n!}{r! (n-r)!} \]

Each term in the expansion is called the \((r+1)^{th}\) term and is expressed as:

\[ T_{r+1} = {n \choose r} a^{n-r} b^r \]

Worked Illustrations

Example 1: Expand \((x + y)^3\).

Solution:

Using the binomial theorem,

\[ (x + y)^3 = \sum_{r=0}^3 {3 \choose r} x^{3-r} y^r \]

Calculating each term:

• For \(r=0\): \({3 \choose 0} x^3 y^0 = 1 \times x^3 = x^3\)
• For \(r=1\): \({3 \choose 1} x^2 y^1 = 3 x^2 y\)
• For \(r=2\): \({3 \choose 2} x^1 y^2 = 3 x y^2\)
• For \(r=3\): \({3 \choose 3} x^0 y^3 = y^3\)

Therefore,

\[ (x + y)^3 = x^3 + 3x^2 y + 3 x y^2 + y^3 \]

Solved Examples

Example 2: Find the middle term(s) in the expansion of \((2 + x)^6\).

Solution:

Since \(n=6\) is even, there is one middle term at position \(\frac{n}{2} + 1 = 4\).

The 4th term is:

\[ T_4 = {6 \choose 3} 2^{6-3} x^3 = {6 \choose 3} 2^3 x^3 \]

Calculate \({6 \choose 3}\):

\[ {6 \choose 3} = \frac{6!}{3!3!} = \frac{720}{6 \times 6} = 20 \]

Calculate \(2^3 = 8\).

Therefore,

\[ T_4 = 20 \times 8 \times x^3 = 160 x^3 \]

The middle term is \(160 x^3\).

Practice Set

Level 1 – Easy

• Expand \((1 + x)^4\).
• Find the coefficient of \(x^2\) in \((3 + x)^5\).
• Write the 3rd term in the expansion of \((x - 2)^5\).

Level 2 – Moderate

• Find the middle term(s) in the expansion of \((1 + 2x)^7\).
• Find the term independent of \(x\) in the expansion of \((x + \frac{1}{x})^6\).
• Prove that the sum of the coefficients in the expansion of \((1 + x)^n\) is \(2^n\).

Level 3 – Challenging

• Find the coefficient of \(x^5\) in the expansion of \((2x - \frac{1}{x^2})^8\).
• Find the term containing \(x^3\) in the expansion of \((1 + 3x)^9\).
• Using binomial theorem, prove that \(\sum_{r=0}^n (-1)^r {n \choose r} = 0\).

Answer Key

Level 1

• \((1 + x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4\)
• Coefficient of \(x^2\) in \((3 + x)^5\) is \({5 \choose 2} 3^{3} = 10 \times 27 = 270\)
• 3rd term in \((x - 2)^5\) is \({5 \choose 2} x^{3} (-2)^2 = 10 x^3 \times 4 = 40 x^3\)

Level 2

• Middle term is the 4th term: \({7 \choose 3} (1)^{4} (2x)^3 = 35 \times 8 x^3 = 280 x^3\)
• Term independent of \(x\) in \((x + \frac{1}{x})^6\) is the term where powers of \(x\) cancel out, i.e., \(6 - 2r = 0 \Rightarrow r=3\). Term: \({6 \choose 3} x^{3} (\frac{1}{x})^{3} = 20\)
• Sum of coefficients in \((1 + x)^n\) is \(\sum_{r=0}^n {n \choose r} = (1 + 1)^n = 2^n\)

Level 3

• Coefficient of \(x^5\) in \((2x - \frac{1}{x^2})^8\): Let term be \(T_{r+1} = {8 \choose r} (2x)^{8-r} (-\frac{1}{x^2})^r\). Power of \(x\) is \(8 - r - 2r = 8 - 3r\). Set \(8 - 3r = 5 \Rightarrow r=1\). Coefficient: \({8 \choose 1} 2^{7} (-1)^1 = 8 \times 128 \times (-1) = -1024\)
• Term containing \(x^3\) in \((1 + 3x)^9\): \(T_{r+1} = {9 \choose r} 1^{9-r} (3x)^r\). Power of \(x\) is \(r\). Set \(r=3\). Term: \({9 \choose 3} 3^3 x^3 = 84 \times 27 x^3 = 2268 x^3\)
• Proof: \(\sum_{r=0}^n (-1)^r {n \choose r} = (1 - 1)^n = 0\)

Quick Reference

Term Number	General Term \(T_{r+1}\)	Binomial Coefficient \({n \choose r}\)
\(r+1\)	\({n \choose r} a^{n-r} b^r\)	\(\frac{n!}{r! (n-r)!}\)

Properties of Binomial Coefficients:

• Sum of coefficients: \(\sum_{r=0}^n {n \choose r} = 2^n\)
• Alternating sum: \(\sum_{r=0}^n (-1)^r {n \choose r} = 0\)
• Sum of even and odd coefficients: \(\sum_{r \text{ even}} {n \choose r} = \sum_{r \text{ odd}} {n \choose r} = 2^{n-1}\)

Glossary

• Binomial Expression: An algebraic expression with two terms connected by + or −.
• Binomial Coefficient \({n \choose r}\): The coefficient of the \(r^{th}\) term in the expansion of \((a+b)^n\).
• General Term: The \((r+1)^{th}\) term in the binomial expansion.
• Middle Term: The term(s) in the middle of the expansion, depending on whether \(n\) is even or odd.
• Independent Term: Term free from variables \(a\) and \(b\) in the expansion.