12.1 Binomial Theorem Expansion

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Use the binomial theorem to expand expressions of the form (a+b)n(a + b)^n, where nn is a positive whole number
  • Simplify the coefficients (the numbers in front of each term) in a binomial expansion
  • Use the formula provided in the formula sheet correctly and confidently

📖 Main Content

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What is a Binomial?

The word binomial simply means "two terms". Any expression that has exactly two parts added or subtracted is a binomial. Examples include:

  • x+5x + 5
  • 2x−3y2x - 3y
  • a+ba + b

When you raise a binomial to a power — for example (a+b)4(a + b)^4 — you need to expand it, meaning you multiply it out and write all the terms. The binomial theorem gives you a quick, reliable method to do this.

Part 1: Pascal's Triangle

Before using the full formula, it helps to understand Pascal's triangle — a number pattern that gives you the coefficients (the numbers in front of each term) for any binomial expansion.

How Pascal's Triangle is Built

  • Every row starts and ends with 1
  • Every other number is found by adding the two numbers directly above it
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
Row 5: 1  5  10  10   5  1
Row 6: 1  6  15  20  15  6  1

How to Use Pascal's Triangle

To expand (a+b)n(a + b)^n:

  1. Find row nn in Pascal's triangle — these are your coefficients
  2. Write the powers of aa starting at nn and decreasing by 1 each term
  3. Write the powers of bb starting at 0 and increasing by 1 each term
  4. The powers of aa and bb in each term must always add up to nn

Important pattern to remember:

  • There are always n+1n + 1 terms in the expansion of (a+b)n(a + b)^n
  • The powers of aa go: n, n−1, n−2, …, 1, 0n,\ n-1,\ n-2,\ \ldots,\ 1,\ 0
  • The powers of bb go: 0, 1, 2, …, n−1, n0,\ 1,\ 2,\ \ldots,\ n-1,\ n

Example 1 — Expanding with Pascal's Triangle

Expand (2+5x)3(2 + 5x)^3

  • The power is 33, so use Row 3 of Pascal's triangle: 1, 3, 3, 1
  • Substitute a=2a = 2 and b=5xb = 5x:
(2+5x)3=1(2)3+3(2)2(5x)+3(2)(5x)2+1(5x)3(2 + 5x)^3 = 1(2)^3 + 3(2)^2(5x) + 3(2)(5x)^2 + 1(5x)^3

Now simplify each term:

  • 1×8=81 \times 8 = 8
  • 3×4×5x=60x3 \times 4 \times 5x = 60x
  • 3×2×25x2=150x23 \times 2 \times 25x^2 = 150x^2
  • 1×125x3=125x31 \times 125x^3 = 125x^3
(2+5x)3=8+60x+150x2+125x3\boxed{(2 + 5x)^3 = 8 + 60x + 150x^2 + 125x^3}

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