Definitions of monomials, binomials, trinomials, degree, leading coefficient, and standard form — with worked examples and practice
A polynomial is an algebraic expression made by adding and subtracting monomials. This guide introduces the basic vocabulary of polynomials: monomials, binomials, and trinomials, degree and leading coefficient, standard form, and how to compare equal polynomials. Examples and four practice sets with full solutions follow.
A monomial is a single algebraic term of the form:
\[ a x^n \]where \( x \) is a variable, \( a \) is a real constant (the coefficient), and \( n \) is a nonnegative integer.
A binomial is the sum or difference of exactly two non-like monomials.
A trinomial is the sum or difference of exactly three non-like monomials.
Why "non-like"? Two terms are like if they have the same variable and the same exponent. For example, \( 2x + 3x \) is not a binomial — the terms combine to \( 5x \), which is a monomial.
A polynomial in \( x \) is the sum of any number of monomials, written in the general form:
\[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]The coefficients \( a_k \) are constants. If the coefficient \( a_n \ne 0 \), then \( n \) (the highest power of the variable) is called the degree of the polynomial, and \( a_n \) is the leading coefficient.
Zero polynomial edge case: the polynomial \( P(x) = 0 \) has all coefficients zero. Its degree is conventionally undefined (or taken as \( -\infty \)), not 0. Every other constant polynomial \( P(x) = c \) with \( c \ne 0 \) has degree 0.
A polynomial is in standard form when its terms are written in descending order by the power of the variable (highest degree first).
Write each polynomial in standard form and determine its degree and leading coefficient.
Two polynomials are equal if and only if they have the same degree and all corresponding coefficients are equal. Write both in standard form first, using zeros for missing terms.
For what values of \( a, b, c \) are \( -x^2 + 4x - 9 \) and \( ax^2 + bx + c \) equal?
Both are already in standard form. Match coefficients term by term:
\[ (-1)x^2 + (4)x + (-9) = (a)x^2 + (b)x + (c) \]So \( a = -1 \), \( b = 4 \), \( c = -9 \).
For what values of \( a, b, c, d \) are \( -x^3 + 4x \) and \( ax^3 + bx + cx^2 + d \) equal?
First, reorder the right side to standard form: \( ax^3 + cx^2 + bx + d \).
Then add zero coefficients to the left side to account for the missing \( x^2 \) and constant terms:
\[ -x^3 + 4x = -1x^3 + 0x^2 + 4x + 0 \]Now match:
\[ -1x^3 + 0x^2 + 4x + 0 = ax^3 + cx^2 + bx + d \]So \( a = -1 \), \( c = 0 \), \( b = 4 \), \( d = 0 \).
| Degree | Name | General form | Examples |
|---|---|---|---|
| 0 | Constant | \( b \) (with \( b \ne 0 \)) | \( -7, 9, -44 \) |
| 1 | Linear | \( ax + b \) | \( -2x + 5, x + 2, x, \tfrac{1}{2}x - 9 \) |
| 2 | Quadratic | \( ax^2 + bx + c \) | \( -2x^2 + 5x - 3, x^2 - 6, x^2 \) |
| 3 | Cubic | \( ax^3 + bx^2 + cx + d \) | \( -x^3 + 5x^2 - 3x + 5, x^3, \tfrac{x^3}{2} \) |
\( 2x^2 \), \( -x^2 - 2x^3 \), \( -6 \), \( -\tfrac{1}{5}x^2 + 4x^5 - 9 \), \( \sqrt{5}\,x^2 - x - 9 \), \( \tfrac{1}{2}x^2 - x^3 + 4 \), \( x^3 + 4 \)
A monomial is a single algebraic term of the form \(a \cdot x^n\) where \(a\) is a real constant and \(n\) is a nonnegative integer. Examples: \(2x^2\), \(-3x\), \(\tfrac{1}{2}x^7\). A plain constant like \(-6\) is also a monomial (the case \(n = 0\)).
The degree is the highest power of the variable that appears with a nonzero coefficient. For example, \(-x^3 + 2x^2 - 1\) has degree 3; \(-x^2 - \tfrac{1}{2}x^3\) has degree 3 after rearranging.
The coefficient of the term with the highest power. For \(-x^3 + 2x^2 - 1\), it's \(-1\) (the coefficient of \(x^3\)).
Standard form means writing the polynomial with terms in descending order of the variable's power. For example, \(4x - 9 + x^2\) becomes \(x^2 + 4x - 9\).
Two polynomials are equal if and only if they have the same degree and all corresponding coefficients are equal. This requires both to be written in standard form first — with zeros filled in for missing terms.
Last updated: October 17, 2026