Polynomials

For example, $x^2$, $2xy$, and $\sqrt{x}$ are the terms of the expression

$$x^2+2xy+\sqrt{x},$$

and the expression

$$2a-ab+c^2$$

is made up of three terms: $2a$, $-ab$, and $c^2$.

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Quick Reference

Concept	Definition	Example
Polynomial	Sum of terms $a{x}^k$, $k$ a nonneg. integer	$3x^2-x+7$
Degree	Largest exponent in the polynomial	$7x^5-x^2+1$ has degree 5
Leading term	Term with the highest exponent	$7x^5$ in the example above
Leading coefficient	Coefficient of the leading term	$7$ in the example above
Monomial	One-term polynomial	$4x^3$
Binomial	Two-term polynomial	$x^2-9$
Trinomial	Three-term polynomial	$x^2+2x+1$
Like terms	Terms with the same variable and exponent	$4x^2$ and $-\sqrt{3}{x}^2$

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Definition of a Polynomial

For example,

$$3,x,7x^{51},-x^3+\frac{1}{\sqrt{14}}x-3,\text{and}7x^5-\pi x^4-\sqrt{2}{x}^3-x^2+x-3$$

are all polynomials. In the last example, the coefficients are $7,-\pi ,-\sqrt{2},-1,1,$ and $-3$.

Note that any constant is also a polynomial because it can be written as $a{x}^0$; for example, $3=3x^0$.

Of course, instead of a polynomial in $x$, we may have polynomials in $y$ or $z$ or any other letter. For example, $4z^3-0.5z+1$ is a polynomial in $z$.

What Is Not a Polynomial?

The rule is simple: negative exponents and fractional exponents are not allowed in a polynomial. Division by a variable (such as $1/x$) is equally forbidden, for the same reason.

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Types of Polynomials by Number of Terms

A polynomial that has only one term is called a monomial. For example, $4x^3$ and $-\sqrt{7}{x}^2$ 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	$4x^3$
Binomial	2	$x^2-9$
Trinomial	3	$x^2+5x+6$
Polynomial	4 or more	$x^3-2x^2+x-5$

Degree of a Polynomial

The largest exponent in a polynomial is called the degree of the polynomial. In $7x^5-\pi x^4-\sqrt{2}{x}^3-x^2+x-3$, the largest power of $x$ is $5$, so the degree of that polynomial is $5$.

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

Name	General form	Degree
Constant polynomial	$a$	$0$
Linear polynomial	$ax+b$	$1$
Quadratic polynomial	$a{x}^2+bx+c$	$2$
Cubic polynomial	$a{x}^3+b{x}^2+cx+d$	$3$

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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

$$-x^2+7x^5+x-\pi x^4-\sqrt{2}{x}^3-3$$

written in standard form is

$$7x^5-\pi x^4-\sqrt{2}{x}^3-x^2+x-3.$$

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 $7x^5$ and the leading coefficient is $7$.

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, $4x^2$, ${\displaystyle \frac{3}{2}}{x}^2$, and $-\sqrt{3}{x}^2$ are like terms, but $3x$ and $3x^2$ are unlike terms.

Like terms can be combined by adding their coefficients:

$$4x^2+\frac{3}{2}{x}^2-\sqrt{3}{x}^2=(4+\frac{3}{2}-\sqrt{3})x^2.$$

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.

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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 $a{x}^k$, where the exponent $k$ is a nonnegative integer and $a$ is any real number. Examples include $x^2-3x+2$ and $5x^3+7$. 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 $4x^3-2x^5+x-7$, the highest exponent is $5$, so the degree is $5$. 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., $6x^2$), a binomial has two terms (e.g., $x+3$), and a trinomial has three terms (e.g., $x^2+5x+6$). 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, $3+x^2-2x$ in standard form is $x^2-2x+3$.

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 $5x^4-3x^2+x-1$, the leading term is $5x^4$ and the leading coefficient is $5$.

Why is $\sqrt{x}+1$ not a polynomial?

Because $\sqrt{x}=x^{1/2}$, and the exponent $\frac{1}{2}$ is not a nonnegative integer. Polynomials allow only whole-number (nonnegative integer) exponents such as $0,1,2,3,\dots$

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 $x^2+3x^2$ instead of $4x^2$) is technically allowed but unusual and may cause confusion. After combining like terms, the polynomial is typically rewritten in standard form.