# Binomial

In algebra, a binomial is a polynomial with two terms[1] —the sum of two monomials—often bound by parentheses or brackets when operated upon. It is the simplest kind of polynomial after the monomials.

## Operations on simple binomials

• The binomial $a^2 - b^2$ can be factored as the product of two other binomials.
$a^2 - b^2 = (a + b)(a - b).$
This is a special case of the more general formula: $a^{n+1} - b^{n+1} = (a - b)\sum_{k=0}^{n} a^{k}\,b^{n-k}$.
This can also be extended to $a^2 + b^2 = a^2 - (ib)^2 = (a - ib)(a + ib)$ when working over the complex numbers
• The product of a pair of linear binomials $(ax+b)$ and $(cx+d)$ is:
$(ax+b)(cx+d) = acx^2+adx+bcx+bd.$
• A binomial raised to the nth power, represented as
$(a + b)^n$
can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. Taking a simple example, the perfect square binomial $(p+q)^2$ can be found by squaring the first term, adding twice the product of the first and second terms and finally adding the square of the second term, to give $p^2+2pq+q^2$.
• A simple but interesting application of the cited binomial formula is the "(m,n)-formula" for generating Pythagorean triples: for m < n, let $a=n^2-m^2$, $b=2mn$, $c=n^2+m^2$, then $a^2+b^2=c^2$.