A polynomial is an algebraic expression built from variables and constants using addition, subtraction and multiplication — where every exponent on a variable is a whole number. Polynomials are usually classified two ways: by how many terms they have, and by their degree.
What counts as a polynomial?
These are polynomials:
5x², 3x + 7, x³ − 2x² + x − 9
These are not polynomials, because of the negative or fractional exponents (or a variable in the denominator):
x⁻², √x (= x^½), 1/x
Classifying by number of terms
| Name | Terms | Example |
|---|---|---|
| Monomial | 1 | 7x² |
| Binomial | 2 | 3x + 4 |
| Trinomial | 3 | x² + 5x + 6 |
Anything with more than three terms is just called a "polynomial" — there are no special one-word names beyond trinomial.
Classifying by degree
The degree is the highest exponent of the variable.
| Name | Degree | Example |
|---|---|---|
| Constant | 0 | 9 |
| Linear | 1 | 2x − 5 |
| Quadratic | 2 | x² + 3x − 1 |
| Cubic | 3 | 4x³ − x |
| Quartic | 4 | x⁴ + 2x² + 1 |
The degree tells you a lot about the polynomial's behaviour — for example, a quadratic graphs as a parabola and can have up to two real roots, which you can find with the quadratic equation calculator.
Key parts of a polynomial
Take 4x³ − 2x² + 7x − 5:
- Terms:
4x³,−2x²,7x,−5 - Coefficients: the numbers
4,−2,7 - Constant term:
−5(no variable) - Leading term:
4x³(highest degree); its coefficient4is the leading coefficient.
Standard form
A polynomial is in standard form when its terms are written from highest degree to lowest:
−2x² + 4x³ + 7 → 4x³ − 2x² + 7
Writing in standard form makes the degree and leading coefficient obvious at a glance, and it's the form most methods (like factoring or the quadratic formula) expect.
Why the classification matters
Knowing the type tells you which tools apply. A linear polynomial solves in one step; a quadratic uses the quadratic formula; higher degrees often need factoring first. Recognising the type is the first move in almost every algebra problem.
Frequently asked questions
What is a polynomial?
What is the degree of a polynomial?
What is the difference between a monomial, binomial and trinomial?
What is a polynomial in standard form?
Is the square root of x a polynomial?
What is the leading coefficient?
CalcSolver Editorial Team
Math & Education Editors
The CalcSolver Editorial Team writes clear, accurate guides on math and calculators. Every article is reviewed for correctness and explained step by step.