Binomial (polynomial)

In algebra, a binomial is a special kind of expression that has just two terms added together. Each of these terms is called a monomial, which means it is a single number or a variable raised to a power, like 5 or x². Binomials are the simplest type of something called a sparse polynomial, which means they have fewer terms than more complex expressions.

Binomials are very important in a part of math called algebraic geometry. They help describe special sets of solutions called toric varieties. These varieties are defined by something called a toric ideal, which is made up of binomials that are differences between monomials. This means each binomial in a toric ideal looks like one monomial minus another, such as x³ - y₂.

When we study these ideals using a method called Buchberger's algorithm, the simplest set of equations that generate the ideal — known as the Gröbner basis — will also only include differences of monomials. This makes working with binomials easier and helps mathematicians understand the structure of these problems better.

Definition

A binomial is a special kind of math expression that adds or subtracts two simple parts called monomials. In the simplest case, with just one variable, it looks like:

a x<sup>m</sup> − b x<sup>n</sup>

Here, a and b are numbers, m and n are different whole numbers that are zero or larger, and x is the variable.

For more complex cases, a binomial can involve many variables, but it always has just two parts that are either added or subtracted.

Examples

Here are some examples of binomials. A binomial is made up of two parts added together. For example:

• 3x - 2x²
• xy + yx²
• 0.9x³ + πy²
• 2x³ + 7
• 11x - 2x²

Each of these has just two terms added or subtracted.

Operations on simple binomials

The binomial (x^2 - y^2), called the difference of two squares, can be factored as ((x - y)(x + y)). This is a special case of a more general formula.

When multiplying two linear binomials ((ax + b)) and ((cx + d)), the result is a trinomial: ((ax + b)(cx + d) = acx^2 + (ad + bc)x + bd).

A binomial raised to the (n)th power, like ((x + y)^n), can be expanded using the binomial theorem or Pascal's triangle. For example, the square ((x + y)^2) equals (x^2 + 2xy + y^2).

Binomials that are sums or differences of cubes can also be factored into smaller-degree polynomials.