The parts of an expression connected by the signs $+$ or $-$ are called the terms of the expression.
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 $c^{2}$.
Quick Reference
| Concept | Definition | Example |
|---|---|---|
| Polynomial | Sum of terms $ax^k$, $k$ 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 $-\sqrt{3}\,x^2$ |
Definition of a Polynomial
A polynomial, or more precisely a polynomial in $x$, is an algebraic expression consisting of terms of the form $ax^{k}$, where $k$ is a nonnegative integer (that is, zero or a natural number $0,1,2,\ldots$) and $a$ is a real number called the coefficient of the term.
For example,
$ 3,\quad x,\quad 7x^{51},\quad -x^{3}+\frac{1}{\sqrt{14}}x-3,\quad\text{and}\quad 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 $ax^{0}$; 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?
Examples of expressions that are not polynomials:
$ 7x^{12}-4x^{-3}+8x-12,\qquad 3x^{2}+\frac{1}{x}-9,\qquad 4\sqrt{x}+5x^{2}. $The first expression is not a polynomial because it contains a negative exponent $(-3)$, while all exponents must be nonnegative integers. The second expression is not a polynomial because $1/x=x^{-1}$, and again all exponents must be nonnegative integers. Similarly, $4\sqrt{x}+5x^{2}$ is not a polynomial because $4\sqrt{x}=4x^{1/2}$ and the exponent $\tfrac{1}{2}$ 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 $1/x$) 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, $4x^{3}$ 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 $a\neq 0$ then:
| Name | General form | Degree |
|---|---|---|
| Constant polynomial | $a$ | $0$ |
| Linear polynomial | $ax+b$ | $1$ |
| Quadratic polynomial | $ax^{2}+bx+c$ | $2$ |
| Cubic polynomial | $ax^{3}+bx^{2}+cx+d$ | $3$ |
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, $4x^{2}$, $\dfrac{3}{2}x^{2}$, 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}+\tfrac{3}{2}x^{2}-\sqrt{3}\,x^{2} = \bigl(4+\tfrac{3}{2}-\sqrt{3}\bigr)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.
Frequently Asked Questions
