Overview
This lecture covers Pascal's Triangle and its application to binomial expansion using the Binomial Theorem, with worked examples.
Building Pascal's Triangle
- Pascal's Triangle starts and ends each row with 1.
- Each inner number is the sum of the two numbers directly above it.
- The rows correspond to the power of the binomial being expanded.
Binomial Expansion using Pascal's Triangle
- The Binomial Theorem uses coefficients from Pascal's Triangle to expand expressions like (a + b)^n.
- Powers of the first term (a) decrease from n to 0; powers of the second term (b) increase from 0 to n.
- The sum of the exponents in each term always equals n.
Example: (x + 5)^4
- Coefficients: 1, 4, 6, 4, 1 (from the 4th row of Pascal's Triangle).
- Terms: x^4, x^35, x^25^2, x*5^3, 5^4.
- Expanded form: x^4 + 20x^3 + 150x^2 + 500x + 625.
Example: (3x - 4y)^5
- Coefficients: 1, 5, 10, 10, 5, 1 (from the 5th row).
- Use 3x as the first term (a), -4y as the second (b).
- Alternate signs based on powers of the negative term.
- Expanded terms switch sign at each power, with calculations for each term.
Key Terms & Definitions
- Pascal's Triangle — A triangular array where each number is the sum of the two above it, used for binomial coefficients.
- Binomial Expansion — Expanding (a + b)^n into a sum using coefficients from Pascal's Triangle.
- Binomial Theorem — The formula that expresses (a + b)^n as a sum of terms with binomial coefficients.
Action Items / Next Steps
- Practice expanding (a + b)^n using Pascal’s Triangle for various n.
- Review homework problems on binomial expansion.
- Prepare questions about negative and variable terms in binomial expansions.