Binomial expansion calculator (a + bx)^n
Enter a, b and a positive integer n and this tool expands (a + bx)^n fully, showing each binomial coefficient nCr, the matching row of Pascal's triangle, and the first four terms in ascending powers of x — the exact form A Level questions ask for.
Expand (a + bx)ⁿ
1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵
First four terms in ascending powers of x: 1 + 10x + 40x² + 80x³
Pascal's triangle row: 1, 5, 10, 10, 5, 1
Term by term
- r = 0: 1
- r = 1: 10x
- r = 2: 40x²
- r = 3: 80x³
- r = 4: 80x⁴
- r = 5: 32x⁵
Working
- Use (a + bx)^n = Σ nCr × a^(n−r) × (bx)^r with a = 1, b = 2 and n = 5.
- Row 5 of Pascal's triangle: 1, 5, 10, 10, 5, 1.
- r = 0: 1 × 1⁵ × (2x)⁰ = 1
- r = 1: 5 × 1⁴ × (2x)¹ = 10x
- r = 2: 10 × 1³ × (2x)² = 40x²
- r = 3: 10 × 1² × (2x)³ = 80x³
The method behind it
- (a + b)^n = sum nCr a^(n-r) b^r with nCr = n! / (r!(n-r)!).
- Powers of a fall from n to 0 while powers of b rise from 0 to n.
- The coefficients are the nth row of Pascal's triangle.
- Ascending powers of x means start with the constant term.
How to use this tool
1. Write the bracket in the form (a + bx)^n
Identify the constant a, the coefficient b of x, and the power n. For (3 - 2x)^5 you would enter a = 3, b = -2 and n = 5.
2. Enter the values
n must be a positive whole number up to 12. a and b can be negative or decimal.
3. Use the terms you need
For a question asking for the first three or four terms in ascending powers of x, read them straight off the ascending list.
Frequently asked questions
What is the binomial expansion formula?
(a + b)^n = sum of nCr x a^(n-r) x b^r for r from 0 to n, where nCr = n! / (r!(n-r)!). Each term drops one power of a and gains one power of b.
How does Pascal's triangle help?
Each row of Pascal's triangle gives the binomial coefficients for that power, so row 5 (1, 5, 10, 10, 5, 1) gives the coefficients of (a + b)^5 without calculating factorials.
How do you find a particular term?
The term in x^r is nCr x a^(n-r) x (bx)^r. Substitute the value of r you need rather than expanding the whole bracket.
Does this work for negative or fractional powers?
Not in this tool. Negative and fractional powers give an infinite series that is only valid for a limited range of x, which is a separate A Level topic.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.