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Flip to reveal answersHow do you expand (a + b)ⁿ?
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All 9 Flashcards — Binomial expansion
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Question
How do you expand (a + b)ⁿ?
Answer
Multiply each coefficient ⁿCᵣ by a falling power of a and a rising power of b: aⁿ + ⁿC₁aⁿ⁻¹b + … + bⁿ.
Question
Expand (x + 3)⁴.
Answer
Coeffs 1, 4, 6, 4, 1 with rising powers of 3: x⁴ + 12x³ + 54x² + 108x + 81.
Question
How do you handle a coefficient like (3x)²?
Answer
Raise the WHOLE term: (3x)² = 9x², not 3x². Always bracket the term before squaring/cubing.
Question
What happens to signs when expanding (a − b)ⁿ?
Answer
Use −b as the second term; even powers come out +, odd powers −. So the signs alternate: + − + − …
Question
Expand (1 − 2x)⁴.
Answer
1 + 4(−2x) + 6(−2x)² + 4(−2x)³ + (−2x)⁴ = 1 − 8x + 24x² − 32x³ + 16x⁴.
Question
How do you find just the first few terms in ascending powers of x?
Answer
Take r = 0, 1, 2, 3 in turn (the lowest powers of x) and stop — no need for the whole expansion.
Question
Find the first three terms, ascending powers, of (1 + x)¹⁰.
Answer
1 + ¹⁰C₁x + ¹⁰C₂x² = 1 + 10x + 45x².
Question
How does binomial expansion link to compound interest?
Answer
(1 + rate)ⁿ is the compound-interest factor; expanding it gives the value (or a quick approximation) by hand.
Question
How can you check a binomial expansion?
Answer
The two powers in every term sum to n, and there are n + 1 terms in total.
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Full study notes for Binomial expansion
Topic 1.9 hub
Binomial theorem
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