The binomial expansion multiplies out (a + b)ⁿ quickly. The coefficients come from Pascal's triangle or from ⁿCᵣ, the number of ways to choose r from n. ⁸C₄ = 70.
(1 + 2x)⁵ = 1 + 5(2x) + 10(2x)² + 10(2x)³ + 5(2x)⁴ + (2x)⁵ = 1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵. To find one term, use ⁿCᵣ aⁿ⁻ʳ bʳ. Keep brackets round terms like (2x) so the powers apply to the number too.
Worked example
Find the coefficient of x² in (1 − 3x)⁵.
- The x² term is ⁵C₂ × (−3x)².
- 10 × 9x² = 90x².
Answer:90
Key words
- binomial
- an expression with two terms
- ⁿCᵣ
- a binomial coefficient
- Pascal's triangle
- a triangle of coefficients
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Common Core-referenced: HSA-APR.C.5