Polynomials

Polynomials constitute what you can regard as a superset that contains other types of expressions, such as monomials, binomials, and trinomials. A monomial consists of a constant coefficient and a single variable. The coefficient must be a real number. The variable often possesses an exponent. If it does, then the exponent must be an integer, and it may not be negative. Here are a few examples of monomials:

5x², 2x², –2, 0, 3x⁴, 2

Monomials are polynomials, as are binomials and trinomials. A binomial consists of two monomials. A trinomial consists of three monomials. A polynomial, generally, can consist of a monomial or a combination of monomials. If you combine monomials to create a polynomial, you use only addition and subtraction to do so. To put it differently, a polynomial provides a sum or difference of monomials, not the quotient or a product. Here are some examples of polynomials:

2x² + 2x, 15a³ + 2a, 15a³, a³, −15a, , 0

Adding or subtracting monomials creates a polynomial. If you multiply or divide monomials, however, you do not create a polynomial. Also, if the variable in an expression contains a negative number, then it is not considered a monomial. Here are a few examples of terms that are not monomials or polynomials:

, ,

In the first expression, 1/x represents a negative exponent (x⁻¹). The second expression contains two monomial expressions (2x and x² + 3), but dividing one by the other does not create a relation based on addition and subtraction. With the third example, the situation is the same. Although the expression includes two monomials, the relation between them is that of division.