The binomial expansion
Multiplying out (a + b)⁷ by hand would take a long time. The binomial expansion gives every term directly, and it lets you pick out one coefficient or make quick approximations.
Part 1 of 3: Learn it
In short
- (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, for r = 0 to n.
- The powers of a go down from n to 0 while the powers of b go up from 0 to n.
- Put any negative sign or number inside b in brackets: (−3x)³ = −27x³.
Where this is in your specification
Spec points: DfE D1 (AQA 7357 D1, Edexcel 9MA0 Pure topic 4, OCR A H240 1.04)
| Board | Topic: Sequences, series and the binomial expansion |
|---|---|
| DfE content | D1-D8 |
| AQA 7357 | D1-D8 |
| Edexcel 9MA0 | Pure topic 4 |
| OCR H240 | 1.04 |
| Higher C847 76 | Algebraic and trigonometric skills: recurrence relations |
Pascal's triangle and nCr
Each row of Pascal's triangle starts and ends with 1, and every other number is the sum of the two above it. Row 4 is 1, 4, 6, 4, 1, which are the coefficients of (a + b)⁴.
For larger n use nCr = n! ÷ (r!(n − r)!), which is the nCr button on a calculator. For example 7C3 = 35.
The expansion
For (1 + x)ⁿ the first few terms are 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + ...
One coefficient and approximations
To find the coefficient of xʳ, write only the term with bʳ in it. To approximate a value such as 1.02⁶, choose x so that (1 + x)ⁿ matches, here x = 0.02, and add the first few terms. Because x is small, its higher powers hardly change the answer.
Work out 5C2.
Show the answer
5! ÷ (2! × 3!) = 10.
Part 2 of 3: See it worked
Worked examples
Example 1
Expand (2 + x)⁴ fully.
- Coefficients 1, 4, 6, 4, 1
- 1(2⁴) + 4(2³)x + 6(2²)x² + 4(2)x³ + 1x⁴
- 16 + 32x + 24x² + 8x³ + x⁴
Answer: 16 + 32x + 24x² + 8x³ + x⁴.
Example 2
Find the coefficient of x³ in the expansion of (1 − 3x)⁷.
- The x³ term is 7C3 × 1⁴ × (−3x)³
- 7C3 = 35 and (−3)³ = −27
- 35 × (−27) = −945
Answer: −945.
Example 3
Use the first three terms of (1 + x)⁶ to estimate 1.02⁶.
- (1 + x)⁶ ≈ 1 + 6x + 15x²
- x = 0.02: 1 + 0.12 + 15 × 0.0004
- = 1 + 0.12 + 0.006 = 1.126
Answer: About 1.126 (the true value is 1.12616...).
Common mistakes
- Forgetting to raise the number in front of x to the power: (2x)³ is 8x³, not 2x³.
- Losing the sign of a negative term.
- Starting the powers of b at 1 instead of 0.
- Giving the whole term when the question asks only for the coefficient.
Write down the first three terms of (1 + 2x)⁵.
Show the answer
1 + 10x + 40x².
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Work out 5C2.
5! ÷ (2! × 3!) = 10.
Write down the first three terms of (1 + 2x)⁵.
1 + 10x + 40x².
What is the coefficient of x² in (3 + x)⁴?
4C2 × 3² = 6 × 9 = 54.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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