6.1 Add and subtract polynomials

Identify polynomials, monomials, binomials and trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form $a{x}^m$ , where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomial are $8,\mathrm{-2}{x}^2,4y^3,\text{and}11z^7$ .

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Here are some examples of polynomials.

$\begin{array}{ccccccccccccc}\text{Polynomial}\hfill & & & b+1\hfill & & & 4y^2-7y+2\hfill & & & 4x^4+x^3+8x^2-9x+1\hfill & & & \\ \text{Monomial}\hfill & & & 14\hfill & & & 8y^2\hfill & & & \mathrm{-9}{x}^3{y}^5\hfill & & & \mathrm{-13}\hfill \\ \text{Binomial}\hfill & & & a+7\hfill & & & 4b-5\hfill & & & y^2-16\hfill & & & 3x^3-9x^2\hfill \\ \text{Trinomial}\hfill & & & x^2-7x+12\hfill & & & 9y^2+2y-8\hfill & & & 6m^4-m^3+8m\hfill & & & z^4+3z^2-1\hfill \end{array}$

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial , binomial , and trinomial when referring to these special polynomials and just call all the rest polynomials .

Determine the degree of polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial    that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.