Polynomials Explained
8 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
A polynomial is an expression made of terms that each have a non-negative whole-number power of a variable, such as 2x³ − 5x + 7. Polynomials describe curves, areas and volumes, and they are the main objects of algebra. Here you will learn the vocabulary and the three basic operations.
What makes an expression a polynomial
Every term must be a number times a variable raised to a whole-number power (0, 1, 2, …). So 4x² + 3x − 9 is a polynomial, but 3/x, √x and 2ˣ are not, because they have a variable in the denominator, under a root or in the exponent.
Naming polynomials
| By number of terms | Example |
|---|---|
| Monomial (1 term) | 7x³ |
| Binomial (2 terms) | x² − 4 |
| Trinomial (3 terms) | 2x² + 3x − 1 |
| By degree | Name | Example |
|---|---|---|
| 1 | Linear | 3x + 2 |
| 2 | Quadratic | x² − 5x + 6 |
| 3 | Cubic | x³ − x |
The degree is the highest power. Standard form lists terms from highest power to lowest.
Adding and subtracting
Combine like terms. (3x² + 2x − 1) + (x² − 4x + 6) = 4x² − 2x + 5. For subtraction, change every sign in the second polynomial first: (3x² + 2x − 1) − (x² − 4x + 6) = 2x² + 6x − 7.
Multiplying
Multiply every term of one polynomial by every term of the other, then combine like terms. (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15. For a larger example, (2x² − 3x + 5)(x + 4) = 2x³ + 5x² − 7x + 20. The degree of the product is the sum of the degrees. Verify any product with the polynomial calculator.
Roots and graphs
A root of a polynomial is a value of x that makes it zero, which is where its graph crosses the x-axis. A polynomial of degree n has at most n real roots. Explore shapes with the graphing calculator: a quadratic is a parabola, a cubic has an S-shape, and higher degrees add more turns.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Multiply (x + 3)(x − 5).
Show answer
Product: x² − 2x − 15
- Find the roots of f(x) = x² − 6x + 8.
Show answer
Function: f(x) = x^2 - 6x + 8
- Solve x² − 7x + 10 = 0.
Show answer
x₁: 5 · x₂: 2
Frequently asked questions
What is the degree of a polynomial?
The highest exponent of the variable, among terms with a non-zero coefficient.
Is a constant a polynomial?
Yes. A non-zero constant has degree 0.
Why is 1/x not a polynomial?
Because its exponent is −1, and polynomial exponents must be non-negative whole numbers.
How many roots can a polynomial have?
A polynomial of degree n has at most n real roots, and exactly n roots if complex ones are counted.
Try it yourself
Polynomial CalculatorAdd, subtract and multiply polynomials and evaluate the result at any x.Graphing CalculatorPlot any function of x, find its roots and y-intercept and see its highest and lowest values.Quadratic Equation SolverSolve ax² + bx + c = 0 with the discriminant, roots and vertex.Keep learning
Algebraic Expressions8 min readFactoring Polynomials9 min readHow to Solve Quadratic Equations8 min readWritten and reviewed by the SolveCalcPro editorial team. Spot a mistake? Tell us. See our editorial policy.