The big idea: To expand a bracket like (x + 3)⁴, multiply each coefficient nCr by a falling power of the first term and a rising power of the second.
IB-style question — expand with a constant
Expand (x + 3)⁴.
Step by step
- Row 4 coefficients are 1, 4, 6, 4, 1; powers of x fall 4 → 0, powers of 3 rise 0 → 4.
- Work out each coefficient (3² = 9, 3³ = 27, 3⁴ = 81).
Final answer
x⁴ + 12x³ + 54x² + 108x + 81.
Quick check: In every term the two powers should sum to n (= 4 here), and there should be n + 1 = 5 terms.
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Raise the whole term: When the second term has a coefficient or a minus sign — like (2 + 3x)³ or (1 − 2x)⁴ — raise the whole term to the power: (3x)² = 9x², (−2x)³ = −8x³.
IB-style question — a coefficient inside
Expand (2 + 3x)³.
Step by step
- Coefficients row 3: 1, 3, 3, 1. Keep the whole 3x together.
- Raise the 3 to the power too: (3x)² = 9x², (3x)³ = 27x³.
Final answer
8 + 36x + 54x² + 27x³.
IB-style question — alternating signs
Expand (1 − 2x)⁴.
Step by step
- Use −2x as the second term; even powers turn it +, odd powers −.
- Compute each: (−2x)² = 4x², (−2x)³ = −8x³, (−2x)⁴ = 16x⁴.
Final answer
1 − 8x + 24x² − 32x³ + 16x⁴.
IB-style question — a minus, in ascending powers
Expand (1 − a)³ in ascending powers of a.
Step by step
- Row 3 coefficients are 1, 3, 3, 1; the second term is −a, so put −a everywhere.
- An even power of −a is +, an odd power is −, so the signs go + − + −.
Final answer
1 − 3a + 3a² − a³ — with a minus inside, the signs simply alternate.
The #1 slip: The coefficient and sign are part of the term: (3x)² = 9x² (not 3x²), and (−2x)³ = −8x³ (not −2x³).
Always put the whole term in a bracket before raising it.
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You don't always need the whole thing: Often you only need the first few terms in ascending powers of x — just take r = 0, 1, 2, 3 in turn, and stop.
IB-style question — first four terms
Find the first four terms, in ascending powers of x, of (1 + x)¹⁰.
Step by step
- Take r = 0, 1, 2, 3 (a = 1, so its powers are all 1).
- Compute the coefficients ¹⁰C₁ = 10, ¹⁰C₂ = 45, ¹⁰C₃ = 120.
Final answer
1 + 10x + 45x² + 120x³ + …
Why this is useful — the finance link: Putting x = a small rate into (1 + x)ⁿ is exactly compound interest — e.g. (1 + 0.05)⁴.
The expansion lets you compute or approximate it by hand.
(See Financial applications.)