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In algebra, a binomial is a polynomial which is the sum of two terms, which are monomials.[1] It is the simplest kind of polynomial after the monomials.

Definition

A binomial is a polynomial, which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form

\( a x^n - bx^m \,, \)

where a and b are numbers, and n and m are distinct nonnegative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable. In some contexts, the exponents m and n may be negative, in which case the monomial is a Laurent binomial.

More generally, a binomial may be written[2] as:

\( a x_1^{n_1}\dotsb x_i^{n_i} - b x_1^{m_1}\dotsb x_i^{m_i} \)

Some examples of binomials are:

\( 3x - 2x^2 \)
\( xy + yx^2 \)
\( x^2 + y^2 \)

Operations on simple binomials

The binomial \(x^2 - y^2 \) can be factored as the product of two other binomials.

\( x^2 - y^2 = (x + y)(x - y). \)

This is a special case of the more general formula:\( x^{n+1} - y^{n+1} = (x - y)\sum_{k=0}^{n} x^{k}\,y^{n-k}.
This can also be extended to \( x^2 + y^2 = x^2 - (iy)^2 = (x - iy)(x + iy) \)when working over the complex numbers

The product of a pair of linear binomials (ax+b) and (cx+d) is a trinomial:

\( (ax+b)(cx+d) = acx^2+(ad+bc)x+bd. \)

A binomial raised to the nth power, represented as

\( (x + y)^n \)

can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square \((x+y)^2 \) of the binomial \(x+y \) is equal to the sum of the squares of the two terms and twice the product of the terms, that is \(x^2+2xy+y^2. The numbers (1,2,1) appearing as multipliers for the terms in this expansion are binomial coefficients two rows down from the top of Pascal's triangle. The expansion of the nth power uses the numbers n rows down from the top of the triangle.

An application of above formula for the square of a binomial is the "(m,n)-formula" for generating Pythagorean triples: for m < n, let \(a=n^2-m^2, b=2mn, c=n^2+m^2, then a^2+b^2=c^2. \)

Binomials that are sums or differences of cubes can be factored into lower-order polynomials as follows:

\( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \)
\( x^3 - y^3 = (x - y)(x^2 + xy + y^2) \)