The Binomial Expansion
Expand (a + b)ⁿ with coefficients from Pascal's triangle or the nCr key, pick out a single term without writing the rest, and use the first few terms as an approximation when one term is small.
What a learner can do afterwards
- Expand (1 + 2x)⁵ in full
- Find the coefficient of a stated power without expanding everything
- Approximate 1.02⁸ from the first three terms and comment on the error
1 · Read
The theorem expands a plus b to the n with no repeated multiplication. Powers of a fall from n to 0 while powers of b climb from 0 to n. Each term carries n choose k. A power of n always gives n plus 1 terms, which catches missing term slips.
Row 5 of the triangle reads 1, 5, 10, 10, 5, 1. Pair each with the right power of 2x to expand 1 plus 2x to the 5. The full form is 1 plus 10x plus 40x squared plus 80x cubed plus 80x to the 4 plus 32x to the 5. Each entry is the sum of the two above it.
Single terms need no full expansion. The term with b to the k carries n choose k. For the x cubed term in 1 plus 2x to the 5, take 10 times 2 cubed. That gives a coefficient of 80. Track the powers of the second term with care.
First terms approximate when one part is small. For 1.02 to the 8, use 1 plus 8 times 0.02 plus 28 times 0.02 squared. That is 1.1712. Dropped terms are tiny but positive, so the estimate sits just below the true value.
Pair each coefficient with falling and rising powers, pluck single terms with n choose k, and approximate with the first terms.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.