Binomial Theorem Summary

Key Concepts

• The binomial theorem provides a method to expand expressions of the form (a + b)ⁿ for positive integral n.
• The expansion is given by:
  • (a + b)ⁿ = Σ (nCk * a^(n-k) * b^k) for k = 0 to n
• The coefficients nCk are known as binomial coefficients and can be found in Pascal's triangle.

Important Observations

• The total number of terms in the expansion is n + 1.
• The powers of a decrease while the powers of b increase in each term.
• The sum of the indices of a and b in each term equals n.

Examples

• Expand (x + 2)⁶:
  • (x + 2)⁶ = 6C0 * x⁶ + 6C1 * x⁵ * 2 + 6C2 * x⁴ * 2² + 6C3 * x³ * 2³ + 6C4 * x² * 2⁴ + 6C5 * x * 2⁵ + 6C6 * 2⁶
  • Result: x⁶ + 12x⁵ + 60x⁴ + 160x³ + 240x² + 192x + 64

Applications

• Used to evaluate powers of numbers that are close to a base (e.g., (98)⁵ = (100 - 2)⁵).
• Can be applied to find approximations and to prove divisibility properties.

Pascal's Triangle

• A triangular array of binomial coefficients:
  • Row 0: 1
  • Row 1: 1, 1
  • Row 2: 1, 2, 1
  • Row 3: 1, 3, 3, 1
  • Row 4: 1, 4, 6, 4, 1

Historical Note

• The concept of binomial coefficients has been known since ancient times, with significant contributions from mathematicians like Blaise Pascal.

Binomial Theorem Notes

Overview

Formula

• $n$ is a positive integer
• ${n \choose k}$ are the binomial coefficients

Pascal's Triangle

• The coefficients of the binomial expansions can be arranged in a triangular format known as Pascal's Triangle.
• Each row corresponds to the coefficients of the expansion of $(a + b)^n$.

Coefficients in Pascal's Triangle

Historical Context

• The concept of binomial coefficients was known to ancient Indian mathematicians and was represented in a diagram called Meru-Prastara.
• The term 'binomial coefficients' was introduced by Michael Stifel in the 16th century.
• Blaise Pascal constructed the triangle in the 17th century, which is now commonly referred to as Pascal's Triangle.

Applications

Examples of Evaluations

1. Evaluate $(96)^3$ using binomial theorem.
2. Evaluate $(102)^5$ using binomial theorem.
3. Determine which is larger: $10000$ or $1000$ using binomial theorem.
4. Approximate $(0.99)^5$ using the first three terms of its expansion.

Miscellaneous Exercises

1. Prove that if $a$ and $b$ are distinct integers, then $a - b$ is a factor of $a^n - b^n$ for positive integers $n$.
2. Expand $(3x^2 + 2ax + 3a^2)^3$ using the binomial theorem.