4.3 Polynomials: Definition, Classification, and Operations

Learning Objectives

After completing this section, you should be able to:

Distinguish between polynomial and non-polynomial expressions.
Identify terms, coefficients, and degree in a polynomial.
Classify a polynomial as a monomial, binomial, or trinomial based on the number of terms.
Distinguish like and unlike terms.
Write a polynomial in standard form (descending powers).
Add and subtract polynomials by combining like terms.
Evaluate a polynomial for a given value of its variable.

What Is a Polynomial?

A polynomial is an algebraic expression that involves variables raised to whole-number exponents and real-number coefficients.

Examples of polynomials:

$3x^2+5x-7$
$-2y^3+y^2-4$
$6a^2b-3ab^2+8$

Not polynomials:

$\frac{1}{x}$ (contains a variable in the denominator)
$\sqrt{x}+4$ (contains a square root of $x$ )
$3x^{-2}+5$ (contains a negative exponent)

Parts of a Polynomial

Take $3x^2+5x-7$ . The parts that are added or subtracted are called terms.

Each term has a number part called the coefficient and a degree (the exponent on the variable).

The degree of the entire polynomial is the highest degree of any term.

**Table 4.3A**
Term	Coefficient	Degree of term	Degree of polynomial
$3x^2$	3	2	2
$5x$	5	1
$-7$	-7	0

Example 4.3.1

Identify the terms, coefficients, and degree for the polynomial $7x^3+2x-5$ .

Solution

Terms: $7x^3$ , $2x$ , $-5$
Coefficients: 7, 2, -5
Degree: 3 (the highest exponent on $x$ )

Example 4.3.2

Identify the terms, coefficients, and degree for the polynomial $5m^3n^2-2m^2n+7$ .

Solution

Terms: $5m^3n^2$ , $-2m^2n$ , $7$
Coefficients: 5, -2, 7
Degree: 5 (add exponents in the highest-degree term: $3+2=5$ )

Try It

Identify the terms, coefficients, and degree.

Monomials, Binomials, and Trinomials

A polynomial is made up of one or more terms.

We can name polynomials according to the number of terms they have.

**Table 4.3B**
Type	Number of Terms	Example	Explanation
Monomial	1 term	$7x^3y^4$	Contains only one term.
Binomial	2 terms	$9a^2+5a$	Contains two terms separated by + or −.
Trinomial	3 terms	$4m^2+7m-8$	Contains three terms separated by + or −.

Note: Terms are separated by plus (+) or minus (−) signs.

Example 4.3.3

Determine whether each expression is a monomial, binomial, or trinomial.

a) $12x^3$

Solution

There is only one term.

Classification: Monomial

b) $3x^2+5x$

Solution

There are two terms separated by a plus sign.

Classification: Binomial

c) $x^2-2x+9$

Solution

There are three terms separated by + and −.

Classification: Trinomial

Example 4.3.4

Polynomials with more than three terms

Sometimes, you’ll see polynomials with four or more terms. These are simply called polynomials (no special name).

Example: $4x^3+3x^2-2x+1$

This expression has four terms → a polynomial with four terms.

Try It

Classify each as a monomial, binomial, or trinomial.

Like Terms vs. Unlike Terms

Like terms have identical variable parts (same letters raised to the same powers).

Only like terms can be combined by adding and subtracting.

Examples

$3x^2$ and $-7x^2$ → like terms
$4y^3$ and $5y^2$ → unlike terms
$-4a^2b$ and $10a^2b$ → like terms

Example 4.3.5

Circle (or list) the like terms.

a) $3x^2,\;5x^2,\;-7x^2,\;4x^3$

Solution

Like terms: $3x^2,\;5x^2,\;-7x^2$

b) $6y^3,\;-2y^2,\;9y^3,\;y^3$

Solution

Like terms: $6y^3,\;9y^3,\;y^3$

Try It

Identify the like terms.

Standard Form

A polynomial is written in standard form when its terms are arranged in descending powers of one variable.

Example 4.3.6

Rewrite in standard form: $2x-7+4x^3$

Solution

$2x-7+4x^3=4x^3+2x-7$

Note: Once written in standard form, the degree is the exponent on the first term.

Example 4.3.7

Rewrite in standard form and state the degree: $y-9y^4+3-5y^2$

Solution

$y-9y^4+3-5y^2=-9y^4-5y^2+y+3$

Degree: 4

Try It

Rewrite in standard form and state the degree.

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Essentials of Mathematics Copyright © 2026 by Scott McDaniel is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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