Binomial Expansion Approximations and Estimations

Examples, solutions, videos, activities, and worksheets that are suitable for A Level Maths.
How to answer questions on Binomial Expansion?

Introduction Infinite series expansion (negative & fractions)
Core 4 - Binomial Expansion (1)

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Examples of typical questions (approximations & partial fractions)
Core 4 - Binomial Expansion (2)

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Binomial Expansion
Edexcel Core Maths C4 June 2010 Q5(b) : ExamSolutions

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Revision of A level binomial expansions - questions and answers 1
A rapid review of the definition of a binomial expansion followed with some illustrations of how and why the coefficients are what they are? Demonstrates that the coefficients have a prescribed pattern.

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Revision of A level binomial expansions - questions and answers 2
This video gives precise definitions or formulae for the coefficients of a binomial expansion.

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Revision of A level binomial expansions - questions and answers 3
This video presents 4 introductory type questions from A level papers and demonstrates a systematic solution methodology for them.
Example:

1. Find the first 3 terms in ascending powers of x of the binomial expansion (3 + 2x)⁵, giving each term in its simplest form.
2. Find the first 3 terms in ascending powers of x of the binomial expansion (2 + x)⁶, giving each term in its simplest form.
3. Find the first 3 terms in ascending powers of x of the binomial expansion (3 - 2x)⁸, giving each term in its simplest form.

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Revision of A level binomial expansions - questions and answers 4
This video presents 4 typical questions from A level papers and demonstrates a systematic solution methodology for solving them.
Example:

1. a) write down the first 3 terms in ascending powers of x of (1 + px)¹², where p is a non-zero constant.
   b) Given that in the expansion, the coefficient of x is (-q) and the coefficient of x² is 11q, find the values of p and q.
2. a) Find the first 4 terms in ascending powers of x of the binomial expansion (1 + dx)¹⁰, where d is a non-zero constant. Give each term in its simplest form.
   b) Given that in the expansion, the coefficient of x³ is double that of x², find the value of d.
3. a) Find the first 4 terms in ascending powers of x of the binomial expansion (1 + kx)⁶, where k is a non-zero constant.
   b) Given that in the expansion, the coefficients of x and x² are equal, find (i) the value of k and (ii) the coefficient of x³.
4. a) Find the first 4 terms in ascending powers of x of the binomial expansion (1 + px)⁹, where p is a non-zero constant.
   b) The first three terms as 1, 36x, qx², where q is constant. Find the values of p and q.

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Revision of A level binomial expansions - questions and answers 5
After a very brief reminder of key formulae which will be used, this video presents 4 less typical questions from A level papers and demonstrates a systematic solution methodology for them. [Minor typo in part b of Q4, should ask for q/p.]
Example:

1. a) Write down the first 4 terms in the expansion of (1 + x/10)¹⁰, giving each term in its simplest form.
   b) Use your expansion to estimate the value of (1.005)¹⁰, giving your answer to 4 decimal places.
2. a) Find the first 4 terms in the expansion of (1 + x/4)⁸, giving each term in its simplest form.
   b) Use your expansion to estimate the value of (1.025)⁸, giving your answer to 4 decimal places.
3. In the binomial expansion of (2 - 5x)²⁰, find an expression for the coefficient of x⁵.
4. b) In the binomial expansion of (1 + x)⁴⁰, the coefficients of x⁴ and x⁵ are p and q respectively. Find the value of q/p.

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