The binomial theorem and Pascal's triangle sheetStructured-answer frameExpand (2x − 1)⁴ using Pascal's triangle, and find the coefficient of x³ in (x − 2)⁵.
How to lay out the answer. Follow the frame, then write it in your own words.
- 1 · Write the row. Row 4: 1 4 6 4 1, with a = 2x and b = −1.
- 2 · Build each term.
- 1(2x)⁴ = 16x⁴
- 4(2x)³(−1) = −32x³
- 6(2x)²(1) = 24x²
- 4(2x)(−1) = −8x
- 1(−1)⁴ = 1
- 3 · Write the expansion. 16x⁴ − 32x³ + 24x² − 8x + 1.
- 4 · Second part: pick the term. x³ needs r = 2 in (x − 2)⁵: 5C2 x³ (−2)² = 10 × 4 = 40.
- 5 · Check the first part with x = 1. (2 − 1)⁴ = 1 and 16 − 32 + 24 − 8 + 1 = 1. ✓
