Identifying 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 \(8,-2{x}^{2},4{y}^{3},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}11{z}^{7}\).

Monomials

A monomial is a term of the form \(a{x}^{m}\), where \(a\) is a constant and \(m\) is a positive whole number.

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.

Polynomials

polynomial—A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

monomial—A polynomial with exactly one term is called a monomial.
binomial—A polynomial with exactly two terms is called a binomial.
trinomial—A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

\(\begin{array}{ccccccccccccc}\text{Polynomial}\hfill & & & \phantom{\rule{2em}{0ex}}b+1\hfill & & & \phantom{\rule{2em}{0ex}}4{y}^{2}-7y+2\hfill & & & \phantom{\rule{2em}{0ex}}4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1\hfill & & & \\ \text{Monomial}\hfill & & & \phantom{\rule{2em}{0ex}}14\hfill & & & \phantom{\rule{2em}{0ex}}8{y}^{2}\hfill & & & \phantom{\rule{2em}{0ex}}-9{x}^{3}{y}^{5}\hfill & & & \phantom{\rule{2em}{0ex}}-13\hfill \\ \text{Binomial}\hfill & & & \phantom{\rule{2em}{0ex}}a+7\hfill & & & \phantom{\rule{2em}{0ex}}4b-5\hfill & & & \phantom{\rule{2em}{0ex}}{y}^{2}-16\hfill & & & \phantom{\rule{2em}{0ex}}3{x}^{3}-9{x}^{2}\hfill \\ \text{Trinomial}\hfill & & & \phantom{\rule{2em}{0ex}}{x}^{2}-7x+12\hfill & & & \phantom{\rule{2em}{0ex}}9{y}^{2}+2y-8\hfill & & & \phantom{\rule{2em}{0ex}}6{m}^{4}-{m}^{3}+8m\hfill & & & \phantom{\rule{2em}{0ex}}{z}^{4}+3{z}^{2}-1\hfill \end{array}\)

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.

Example

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

\(4{y}^{2}-8y-6\)
\(-5{a}^{4}{b}^{2}\)
\(2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4\)
\(13-5{m}^{3}\)
\(q\)

Solution

\(\begin{array}{cccccccccc}& & & \mathbf{\text{Polynomial}}\hfill & & & \mathbf{\text{Number of terms}}\hfill & & & \mathbf{\text{Type}}\hfill \\ \text{(a)}\hfill & & & 4{y}^{2}-8y-6\hfill & & & \hfill 3\hfill & & & \text{Trinomial}\hfill \\ \text{(b)}\hfill & & & -5{a}^{4}{b}^{2}\hfill & & & \hfill 1\hfill & & & \text{Monomial}\hfill \\ \text{(c)}\hfill & & & 2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4\hfill & & & \hfill 5\hfill & & & \text{Polynomial}\hfill \\ \text{(d)}\hfill & & & 13-5{m}^{3}\hfill & & & \hfill 2\hfill & & & \text{Binomial}\hfill \\ \text{(e)}\hfill & & & q\hfill & & & \hfill 1\hfill & & & \text{Monomial}\hfill \end{array}\)

Identifying Polynomials, Monomials, Binomials and Trinomials