Binomial applications
Objective. Use selected terms and coefficients in approximation and proof.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Binomial expansions can approximate numerical values or extract a single coefficient without expanding every term, using the general term C(n,k)aⁿ⁻ᵏ(bx)ᵏ for the term in a chosen power of x. For small x, truncating an expansion after the first few terms gives a close numerical approximation, since higher powers of small x become negligible. For instance, (1+0.01)⁵≈1+5(0.01)=1.05 using just the first two terms of the expansion. In proof-style questions, comparing a specific coefficient across two expressions can establish unknown constants within a binomial expression.
Worked example
Find the coefficient of x³ in the binomial expansion of (1+2x)⁵.
- 1Identify the general term C(5,k)(2x)^k with k=3
- 2Compute C(5,3)=10
- 3Compute 2³=8
- 4Multiply coefficients: 10×8=80
Answer: 80
Concept mastery
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Practice
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Start practisingCommon mistake. Students include every term when only one coefficient is needed, wasting time and risking arithmetic slips; isolate the required term using k directly in C(n,k)aⁿ⁻ᵏ(bx)ᵏ instead of expanding fully.
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