Binomial expansion and valid approximations
| English | 中文 | Pinyin |
|---|---|---|
| binomial coefficient/baɪˈnəʊmɪəl ˌkəʊɪˈfɪʃənt/ | 二项式系数 | èr xiàng shì xì shù |
How does a small change affect a power?
- A small measurement change affects a power of a quantity. An expansion can show the size of first and second effects.
- This lesson studies binomial coefficient 二项式系数: The number of ways to choose a specified number of objects from a set.
Choose the mathematical structure
- For positive integer n, (a+b)^n is a finite binomial expansion. For noninteger n, expand (1+x)^n as 1+nx+n(n-1)x²/2+... with |x|<1. Factor out constants before using this form.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines binomial coefficient?
The number of ways to choose a specified number of objects from a set.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
(1+2x)^5=1+10x+40x²+80x³+80x⁴+32x⁵. For (1+x)^(-1), the first three terms are 1-x+x²; at x=0.1 this gives 0.91 versus the exact 1/1.1≈0.909091.
Binomial expansion and valid approximations
For positive integer n, (a+b)^n is a finite binomial expansion
Compare the model with the worked case and explain one change.
Find the coefficient of x² in (1+2x)^5.
Choose two of five factors and include 2²: C(5,2)×4=40.
Test a tempting shortcut
- An expansion in 2x requires |2x|<1 for the infinite series, not merely |x|<1. Positive-integer expansions are finite and do not have that convergence restriction.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every binomial expansion is a finite polynomial, including powers that are not positive integers. This claim is false. Explain which definition or assumption it violates.
Use 1-x+x² to estimate 1/(1+x) at x=0.1.
Substitute 0.1: 1-0.1+0.01=0.91.
Every binomial expansion is a finite polynomial, including powers that are not positive integers.
An expansion in 2x requires |2x|<1 for the infinite series, not merely |x|<1. Positive-integer expansions are finite and do not have that convergence restriction.
Interpret a new situation
- State the range of validity alongside an approximation. Retain enough terms to support the requested accuracy, and distinguish a coefficient from the whole term containing x.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many terms are in the full expansion of (a+b)^5?
The powers run from 0 through 5, giving six terms.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The number of ways to choose a specified number of objects from a set. Choose the relationship, show the method, check its assumptions and interpret the result.