The binomial expansion multiplies out (a + bx)ⁿ for a positive whole number n without expanding term by term. The coefficients are the binomial coefficients nCr, which form row n of Pascal's triangle, and the general term is nCr × aⁿ⁻ʳ × (bx)ʳ. You can find a single term, compare coefficients to find an unknown, expand a product, and estimate powers such as 0.98⁶.
The binomial expansion
- (a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + … + bⁿ
- (1 + x)ⁿ = 1 + nx + n(n − 1)/2! x² + n(n − 1)(n − 2)/3! x³ + …
- General term: nCr × aⁿ⁻ʳ × (bx)ʳ
- nCr = n! / (r!(n − r)!)
Pascal's triangle and nCr
For (1 + x)ⁿ, the coefficients are the numbers in row n of Pascal's triangle, where each row starts and ends with 1 and each inside number is the sum of the two above it. The same numbers are the binomial coefficients nCr, which your calculator can find directly. The powers of x go up from 0 to n.
Expanding (a + bx)ⁿ
Each term is nCr × aⁿ⁻ʳ × (bx)ʳ, so both a and b are raised to powers. Put the whole of bx in brackets before raising it to a power: (−2x)³ = −8x³. A negative b makes the signs alternate. Fractions are fine for a and b; it is the power n that must be a positive whole number here.
Finding a specific term
You do not need the whole expansion to find one coefficient. The term in xʳ is nCr × aⁿ⁻ʳ × bʳ × xʳ, so choose r to match the power you want and evaluate that single term.
Comparing coefficients
If the question links two coefficients, for example “the coefficient of x² is twice the coefficient of x”, write each coefficient in terms of the unknown, set up the relationship and solve. Cancel common factors first; the result is often a simple linear or quadratic equation.
Products and estimates
To find a coefficient in a product such as (c + dx)(a + bx)ⁿ, expand only as far as you need, then multiply. The x² term comes from c × (the x² coefficient) plus d × (the x coefficient). To estimate a power such as 0.98⁶, choose x so that the bracket equals the number you want, then substitute into the first few terms.
Worked examples
Example 1. Expand (1 + x)⁵ fully.
- Row 5 of Pascal's triangle: 1, 5, 10, 10, 5, 1.
- Attach increasing powers of x.
Answer: 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵.
Marks: M1 for the coefficients, A1 for the expansion.
Example 2. Find the first three terms, in ascending powers of x, of (4 − x)⁵.
- First term: 4⁵ = 1024.
- x term: 5C1 × 4⁴ × (−1) = −1280.
- x² term: 5C2 × 4³ × (−1)² = 10 × 64 = 640.
Answer: 1024 − 1280x + 640x².
Marks: M1 for the general term, A1 for −1280x, A1 for 640x².
Example 3. Find the coefficient of x⁴ in (5 + 2x)⁶.
- The x⁴ term uses r = 4: 6C4 × 5² × 2⁴.
- 15 × 25 × 16.
Answer: 6000.
Marks: M1 for the correct term, A1 for 6000.
Example 4. In (4 + ax)⁵ the coefficient of x² is twice the coefficient of x. Find a, where a ≠ 0.
- Coefficient of x: 5C1 × 4⁴ × a = 1280a. Coefficient of x²: 5C2 × 4³ × a² = 640a².
- 640a² = 2 × 1280a, so 640a(a − 4) = 0.
- a ≠ 0.
Answer: a = 4.
Marks: M1 for the equation, A1 for a = 4.
Example 5. (a) Find the first three terms of (2 + 3x)⁵. (b) Hence find the coefficient of x² in (4 − x)(2 + 3x)⁵.
- (a) 2⁵ = 32; 5 × 2⁴ × 3 = 240; 10 × 2³ × 9 = 720, so 32 + 240x + 720x² + …
- (b) x² terms: 4 × 720x² and (−x) × 240x.
- 4 × 720 − 240 = 2640.
Answer: (a) 32 + 240x + 720x². (b) 2640.
Marks: B1 for (a), M1 for combining, A1 for 2640.
Example 6. (a) Find the first three terms of (1 − 2x)⁶. (b) By substituting x = 0.01, estimate 0.98⁶.
- (a) 1 + 6(−2x) + 15(−2x)² = 1 − 12x + 60x².
- (b) 1 − 2(0.01) = 0.98, so substitute x = 0.01.
- 1 − 0.12 + 0.006 = 0.886.
Answer: 0.98⁶ ≈ 0.886.
Marks: B1 for the expansion, M1 for substituting, A1 for 0.886.
Common mistakes
- Not bracketing bx. (2x)³ = 8x³, not 2x³.
- Forgetting to raise a to a power. In (4 − x)⁵, the first term is 4⁵, not 4.
- Choosing the wrong r. The term in xʳ uses nCr; the x⁴ term has r = 4.
- Losing the sign. With a negative b, odd powers give negative terms.
- Missing a product contribution. In (c + dx)(a + bx)ⁿ, the x² coefficient has two parts.
How the binomial expansion is examined (positive n)
The binomial expansion for positive whole n is a Pure 2 (WMA12) chapter in Edexcel International A-Level. In Edexcel A-Level (9MA0) it is taught in Year 1. Questions typically:
- ask for the first three or four terms in ascending powers of x
- ask for a single coefficient
- give a relationship between coefficients and ask for an unknown
- use the expansion to estimate a value, or extend it to a product
The Pure 2 binomial expansion exam questions include all of these, with mark schemes. Use the binomial expansion worksheet or the A-Level sequences and series revision engine for more.
In Pure 4 the expansion extends to negative and fractional powers: see the binomial expansion for any rational n.
Frequently asked questions
What is the binomial expansion?
A formula for expanding (a + b)ⁿ: the terms are nCr × aⁿ⁻ʳ × bʳ for r = 0 to n.
How do you find nCr?
Use the nCr button on your calculator, or n!/(r!(n − r)!), or read it from row n of Pascal's triangle.
How many terms are in the expansion of (a + b)ⁿ?
n + 1 terms, for a positive whole number n.