The binomial theorem
Each term of (ax + b)^n takes a power of ax and the remaining power of b, with a binomial coefficient in front.
The coefficient of the term with x to the power k is n choose k, multiplied by a to the k and b to the n minus k.
Worked example
(2x + 3)^5 expands to 32x⁵ + 240x⁴ + 720x³ + 1080x² + 810x + 243.
The coefficients are 32, 240, 720, 1080, 810 and 243, and the Pascal row is 1, 5, 10, 10, 5, 1.
Checking an expansion
Put x = 1 into both sides. The bracket becomes a + b and the expansion must add up to (a + b)^n.
For (2x + 3)^5 that is 5⁵ = 3125, and the six coefficients add to 3125, so the expansion is right.
Signs and the constant term
A negative constant alternates the signs. (x − 2)³ gives x³ − 6x² + 12x − 8.
The constant term is always b to the power n, because it is the term with no x in it.
Inputs for the binomial expansion calculator
Coefficient of x
The number multiplying x inside the bracket.
Constant
The number on its own inside the bracket. A negative value alternates the signs.
Power
How many times the bracket multiplies itself, from 0 to 15.
CALCULATOR QUESTIONS
Binomial expansion calculator questions
How do I expand (2x + 3)^5?
Use the binomial theorem with the Pascal row 1, 5, 10, 10, 5, 1. The answer is 32x⁵ + 240x⁴ + 720x³ + 1080x² + 810x + 243.
What is Pascal's triangle used for?
Its rows give the binomial coefficients. Row 5 is 1, 5, 10, 10, 5, 1, which are the coefficients of (a + b)^5.
Why does the constant term have no x?
It is the term where every factor contributed b rather than ax, so it is b to the power n.
Can the power be negative or a fraction?
Not here. This page handles whole-number powers, which give a finite number of terms.