Binomial Theorem Calculator
Calculate with the binomial theorem. This expands the term (a+b)n, the polynom with its individual summands with be displayed. For a and b, other terms can be entered, which will appear in the output. Please enter for n an integer between 2 and 100.
If ax by is selected, the powers are written with superscript numbers, if a^x b^y is selected, ^ is used as the exponentiation symbol. The first notation is easier to read, but the second notation is better for copying.
| The formula is |  |
The binomial coefficients 
calculate as
Examples: n=2 gives the most basic formula (a+b)² = a² + 2 a b + b²
for n=4 with the notation a^x b^y the polynomial a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 results.
With a and -b we get for n=3 (a+(-b))³ = a^3 + 3 a^2 (-b) + 3 a (-b)^2 + (-b)^3
The binomial theorem is an extension of the binomial formulas to general natural number powers. Binomial means two-named. The two names refer to a and b, or whatever terms you want to use here, as long as they are two different ones.
The binomial theorem for the case n=2, i.e. the first binomial formula, was already known to Euclid in the fourth century BC. In the year 510 AD, the Indian mathematician Aryabhata gave a method for solving cube roots, which suggests that he knew the binomial theorem for n=3. The complete binomial theorem was probably first written down by the Persian mathematician Al-Karaji around the year 1000. This document is now lost, but some later writings refer to it. Similar discoveries were also made in China around this time. From Persia, this knowledge, along with many other mathematical discoveries, initially only slowly reached Europe in the late Middle Ages and early modern period.