Binomial expansion and valid approximations
Введено| English | Значение |
|---|---|
| binomial coefficient/baɪˈnəʊmɪəl ˌkəʊɪˈfɪʃənt/ | биномиальный коэффициент |
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.
The method
- 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.
For the infinite expansion of (1+2x)^(-1), which condition is required?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
The validity condition |u|<1 applies to the whole bracket term u=2x, so |x|<1/2. Using |x|<1 would wrongly accept x=0.9, where the series diverges.
A worked case
(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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Choose two of five factors and include 2²: C(5,2)×4=40.
Where it goes wrong
- 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.
Not true: Every binomial expansion is a finite polynomial, including powers that are not positive integers. For a negative or fractional power the expansion is an infinite series, valid only for |x|<1.
Use 1-x+x² to estimate 1/(1+x) at x=0.1.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Substitute 0.1: 1-0.1+0.01=0.91.
For a negative or fractional power the expansion is an infinite series, valid only for |x|<1.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
True. For a negative or fractional power the expansion is an infinite series, valid only for |x|<1.
Applying it
- 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.
How many terms are in the full expansion of (a+b)^5?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The powers run from 0 through 5, giving six terms.
Match each expansion fact to its statement.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Positive-integer powers expand finitely with binomial coefficients; other powers give infinite series that converge only inside the validity interval.
In your course
- Pearson Edexcel International A Level Pure Mathematics, unit P2.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Binomial coefficient: The number of ways to choose a specified number of objects from a set.
- For positive integer n, (a+b)^n is a finite binomial expansion.
- Watch for: an expansion in 2x requires |2x|<1 for the infinite series, not merely |x|<1.
Write from a coefficient of 12
- Expand the idea of (1 plus 3x) to the power 4.
- Find the coefficient of x.
In the expansion of (1 plus 3x) to the power 4, what is the coefficient of x?
4 times 3 is 12.
What is the coefficient of x?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Four ways to pick one factor of 3x give 12. Other options — 40 (the lesson coefficient of x squared); 3 (the number inside the bracket).