Binomial Expansion Overview

Overview

This lecture covers the binomial expansion, including patterns using Pascal's Triangle, calculating coefficients with nCr, and solving both standard and estimation problems involving binomial expressions.

Binomial Expansion Basics

• Binomial expansion is used to expand expressions of the form (a + b)^n for any integer n.
• At GCSE level, expanding small powers (like (x+5)^2) reveals patterns in coefficients.
• For higher powers, patterns in coefficients relate directly to rows of Pascal’s Triangle.
• Each row of Pascal’s Triangle gives the coefficients for the expansion of (a + b)^n.

Pascal’s Triangle & Calculating Coefficients

• Pascal’s Triangle is created by adding numbers diagonally: start with 1, then each new entry is the sum of the two directly above it.
• Rows are: n=0: 1; n=1: 1 1; n=2: 1 2 1; n=3: 1 3 3 1, etc.
• Coefficients can also be calculated with the nCr (“n choose r”) function on a calculator (usually behind the divide button).

Using Binomial Expansion

• For (a + b)^n, each term is: nCr × a^(n-r) × b^r, where r goes from 0 to n.
• In expansions, decrease the power of a and increase the power of b across terms.
• Negative and fractional terms should be carefully handled with correct signs and simplification.

Worked Examples

Example: (x + 3)^3

• Coefficients from 3rd row: 1, 3, 3, 1.
• Expanded: x^3 + 9x^2 + 27x + 27.

Example: (3 - (1/3)x)^5, first four terms

• Use nCr to determine coefficients: 1, 5, 10, 10.
• Substitute values and simplify each term with care to signs and fractions:
  • 243 - 135x + 30x^2 - (10/3)x^3.

Example: (1 + px)^10, first three terms

• First three coefficients: 1, 10, 45.
• Expansion: 1 + 10px + 45p^2x^2.
• If x^2 coefficient = 9 × x coefficient, set 45p^2 = 9 × 10p → p = 2.
• Coefficient of x^2: 45 × 4 = 180.

Example: Estimation with Binomial Expansion

• For (1 - x/4)^10, expand and substitute x as required for approximation (e.g. estimate 0.975^10).
• Use expansion to substitute x = 0.1 and calculate the estimate, rounding as necessary.

Key Terms & Definitions

• Binomial Expansion — Expanding (a + b)^n into a sum of terms using powers of a and b.
• Pascal’s Triangle — A triangular array where each row gives coefficients for the binomial expansion.
• nCr (Combination) — The number of ways to choose r objects from n, used for coefficients.
• Coefficient — The numerical part multiplying variables in each term of the expansion.

Action Items / Next Steps

• Practice expanding binomial expressions using both Pascal’s Triangle and the nCr calculator function.
• Complete any assigned binomial expansion problems, including estimation questions.
• Review and memorize the first several rows of Pascal’s Triangle for quick reference.