Polynomials

The parts of an expression connected by the signs or are called the terms of the expression.

For example, , , and are the terms of the expression

and the expression

is made up of three terms: , , and .

Quick Reference

Concept	Definition	Example
Polynomial	Sum of terms , a nonneg. integer
Degree	Largest exponent in the polynomial	has degree 5
Leading term	Term with the highest exponent	in the example above
Leading coefficient	Coefficient of the leading term	in the example above
Monomial	One-term polynomial
Binomial	Two-term polynomial
Trinomial	Three-term polynomial
Like terms	Terms with the same variable and exponent	and

Definition of a Polynomial

A polynomial, or more precisely a polynomial in , is an algebraic expression consisting of terms of the form , where is a nonnegative integer (that is, zero or a natural number ) and is a real number called the coefficient of the term.

For example,

are all polynomials. In the last example, the coefficients are and .

Note that any constant is also a polynomial because it can be written as ; for example, .

Of course, instead of a polynomial in , we may have polynomials in or or any other letter. For example, is a polynomial in .

What Is Not a Polynomial?

Examples of expressions that are not polynomials:

The first expression is not a polynomial because it contains a negative exponent , while all exponents must be nonnegative integers. The second expression is not a polynomial because , and again all exponents must be nonnegative integers. Similarly, is not a polynomial because and the exponent is not an integer.

The rule is simple: negative exponents and fractional exponents are not allowed in a polynomial. Division by a variable (such as ) is equally forbidden, for the same reason.

Types of Polynomials by Number of Terms

A polynomial that has only one term is called a monomial. For example, and are monomials. A polynomial that has exactly two terms is called a binomial, and a polynomial that has exactly three terms is called a trinomial.

Name	Number of terms	Example
Monomial	1
Binomial	2
Trinomial	3
Polynomial	4 or more

Degree of a Polynomial

The largest exponent in a polynomial is called the degree of the polynomial. In , the largest power of is , so the degree of that polynomial is .

Polynomials of degree 0, 1, 2, and 3 have special names. If then:

Name	General form	Degree
Constant polynomial
Linear polynomial
Quadratic polynomial
Cubic polynomial

Standard Form and Leading Term

A polynomial is written in standard form when its terms are arranged in descending order of their exponents. For example, the polynomial

written in standard form is

The first term of a polynomial in standard form is called the leading term, and its coefficient is called the leading coefficient. In the example above, the leading term is and the leading coefficient is .

Writing a polynomial in standard form makes it easy to read off the degree and the leading coefficient at a glance.

Like Terms

When two or more terms differ only in their numerical coefficients, we say they are similar or like terms. For example, , , and are like terms, but and are unlike terms.

Like terms can be combined by adding their coefficients:

Simplifying a polynomial by combining all like terms and then writing the result in standard form is the usual first step before doing any further algebra.

Frequently Asked Questions

What is a polynomial in simple terms?

A polynomial is an algebraic expression built by adding or subtracting terms of the form

, where the exponent

is a nonnegative integer and

is any real number. Examples include

and

. Expressions with negative or fractional exponents, or with a variable in the denominator, are not polynomials.

How do you find the degree of a polynomial?

Identify the term with the highest exponent. That exponent is the degree. For example, in

, the highest exponent is

, so the degree is

. If you are not sure which exponent is largest, rewrite the polynomial in standard form (descending order) first.

What is the difference between a monomial, a binomial, and a trinomial?

The names refer to the number of terms. A monomial has one term (e.g.,

), a binomial has two terms (e.g.,

), and a trinomial has three terms (e.g.,

). All three are special cases of polynomials.

What is the standard form of a polynomial?

A polynomial is in standard form when its terms are written in descending order of the exponent: highest exponent first, then the next highest, and so on down to the constant term. For example,

in standard form is

What is the leading coefficient of a polynomial?

The leading coefficient is the coefficient of the term with the highest exponent (the leading term) when the polynomial is written in standard form. For

, the leading term is

and the leading coefficient is

Why is

not a polynomial?

Because

, and the exponent

is not a nonnegative integer. Polynomials allow only whole-number (nonnegative integer) exponents such as

Are like terms always combined when writing a polynomial?

By convention, a polynomial is fully simplified when all like terms have been combined. Leaving like terms uncombined (e.g., writing

instead of

) is technically allowed but unusual and may cause confusion. After combining like terms, the polynomial is typically rewritten in standard form.