To expand (a + b)ⁿ, write the binomial coefficients for power n, then give each term a falling power of a and a rising power of b. The skill appears whenever a question says “expand” or “find the first three terms”, and it feeds every other lesson in binomial expansion.
How does the pattern work?
Each term has the form (coefficient) × aⁿ⁻ʳ × bʳ, where r runs from 0 to n. The powers of a and b always add up to n.
For power 4 the coefficients are 1, 4, 6, 4, 1. They come from Pascal’s triangle, where each number is the sum of the two above it, or from the nCr function on your calculator. The row for power n has n + 1 numbers and they add up to 2ⁿ.
A layout that keeps every term in place
- Write the coefficients for the power in a row.
- Write the first bracket part with its powers falling from n to 0, keeping it in brackets: (2x)⁴, (2x)³, (2x)², (2x)¹, (2x)⁰.
- Write the second part with its powers rising from 0 to n, also in brackets: 3⁰, 3¹, 3², 3³, 3⁴.
- Multiply each column together, simplifying the numbers as you go.
- Add the terms and check the result.
Worked example
Expand (2x + 3)⁴.
Step 1, coefficients: 1, 4, 6, 4, 1.
Step 2, term by term:
- 1 × (2x)⁴ × 3⁰ = 1 × 16x⁴ × 1 = 16x⁴
- 4 × (2x)³ × 3¹ = 4 × 8x³ × 3 = 96x³
- 6 × (2x)² × 3² = 6 × 4x² × 9 = 216x²
- 4 × (2x)¹ × 3³ = 4 × 2x × 27 = 216x
- 1 × (2x)⁰ × 3⁴ = 1 × 1 × 81 = 81
Step 3, add:
(2x + 3)⁴ = 16x⁴ + 96x³ + 216x² + 216x + 81
Step 4, check with x = 1: the left side is 5⁴ = 625. The right side is 16 + 96 + 216 + 216 + 81 = 625. They agree.
The mistake to watch for
The most common slip is to raise only the x, not the whole first part.
Mistaken working: 4 × 2x³ × 3 = 24x³
The student wrote 2x³ instead of (2x)³ = 8x³, so the coefficient of x³ came out as 24 instead of 96.
The correction is to keep every part in brackets until you have simplified it. Write (2x)³ first, then 8x³. The check at x = 1 would have caught this at once, because the total would have been far too small.
Check yourself
Try these, then open each answer.
1. Expand (x + 2)⁵.
Show answer
Coefficients 1, 5, 10, 10, 5, 1. Terms: x⁵, 5 × x⁴ × 2 = 10x⁴, 10 × x³ × 4 = 40x³, 10 × x² × 8 = 80x², 5 × x × 16 = 80x, and 2⁵ = 32.
x⁵ + 10x⁴ + 40x³ + 80x² + 80x + 32
Check at x = 1: 3⁵ = 243, and 1 + 10 + 40 + 80 + 80 + 32 = 243.
2. Write the row of binomial coefficients for power 6 and show that it adds up to 64.
Show answer
1, 6, 15, 20, 15, 6, 1. The sum is 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 = 2⁶.
3. Expand (1 + 3x)³.
Show answer
Coefficients 1, 3, 3, 1. Terms: 1, 3 × 3x = 9x, 3 × (3x)² = 3 × 9x² = 27x², and (3x)³ = 27x³.
1 + 9x + 27x² + 27x³
Check at x = 1: 4³ = 64, and 1 + 9 + 27 + 27 = 64.
Where this leads next
Once a full expansion feels reliable, learn to find a specified term without writing every term. Later, checking an expansion using a simple substitution gives you a habit that protects every answer. The non-calculator working trainer is useful for practising exact arithmetic in these coefficients.
Some students can follow each line but lose marks when the layout is left to memory. Our teachers look at exactly that in online one-to-one Additional Mathematics tuition.