Quick answer
Polynomials are entered as coefficient lists from the highest power: [1, 2] is x + 2 and [3, −1] is 3x − 1. Adding them gives [4, 1] = 4x + 1, which equals 9 at x = 2 and has degree 1.
Enter two coefficient lists, an operation and optionally an x value. The calculator returns the result coefficients, the value and the degree.
Related tools: Polynomial Long Division Calculator: Quotient and remainder of one polynomial divided by another. Slope Calculator: Slope, angle, distance and line equation through two points. LCM Calculator: Least common multiple of two or more numbers.
How to use the Polynomial Calculator
- Type first polynomial coefficients (highest power first), for example 1, 2.
- Type second polynomial coefficients, for example 3, -1.
- Choose operation: Add, Subtract and Multiply.
- Optionally, enter evaluate at x (optional).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| First polynomial coefficients (highest power first) | Text, such as 1, 2 | — |
| Second polynomial coefficients | Text, such as 3, -1 | — |
| Operation | Choose one: Add; Subtract; Multiply | Starts at Add |
| Evaluate at x (optional) (optional) | A number | — |
Formula and method
Formula
coefficient-wise arithmetic; multiplication is convolution
IEEE-754 double precision; shown to the decimals indicated.
Coefficients are listed from the highest power down.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: First polynomial coefficients (highest power first) = 1, 2; Second polynomial coefficients = 3, -1; Operation = Add; Evaluate at x (optional) = 2.
| Result | Value |
|---|---|
| Result coefficients | 4, 1 |
| Value at x | 9 |
| Degree | 1 |
Example 2
Inputs: First polynomial coefficients (highest power first) = 2, 0, 1; Second polynomial coefficients = 1, 1; Operation = Subtract.
| Result | Value |
|---|---|
| Result coefficients | 2, -1, 0 |
| Value at x | — |
| Degree | 2 |
Example 3
Inputs: First polynomial coefficients (highest power first) = 1, 1; Second polynomial coefficients = 1, -1; Operation = Multiply; Evaluate at x (optional) = 3.
| Result | Value |
|---|---|
| Result coefficients | 1, 0, -1 |
| Value at x | 8 |
| Degree | 2 |
The first line is the headline result (result coefficients). Below it the polynomial Calculator lists value at x and degree, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Coefficient arithmetic
Addition and subtraction work term by term; multiplication is a convolution: (x + 2)(3x − 1) = 3x² + 5x − 2, i.e. [3, 5, −2]. The calculator works with coefficient lists, not typed expressions.
Evaluating
Evaluating at x = 2: 4 × 2 + 1 = 9. Horner's method evaluates efficiently by nesting. For division use the Polynomial Long Division calculator.
How the result changes with evaluate at x
Holding every other input at the first worked example, this table changes only evaluate at x. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Evaluate at x | Result coefficients | Value at x | Degree |
|---|---|---|---|
| 1 | 4, 1 | 5 | 1 |
| 2 | 4, 1 | 9 | 1 |
| 4 | 4, 1 | 17 | 1 |
| 8 | 4, 1 | 33 | 1 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- First polynomial coefficients (highest power first): left empty shows "Enter first polynomial coefficients (highest power first)." beside that field and no result.
- Second polynomial coefficients: entering "x" shows "“x” is not a number." beside that field and no result.
Choosing between the Polynomial Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Polynomial Long Division Calculator](/math/polynomial-division/) | dividend coefficients and divisor coefficients | Quotient |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
| [Quadratic Formula Calculator](/math/quadratic-formula-calculator/) | a, b and c | Roots |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Coefficient lists, not expressions
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for first polynomial coefficients (highest power first) and second polynomial coefficients to confirm this.
Frequently asked questions
What is the degree?
The highest power with a non-zero coefficient.
How do I enter x²?
Write the coefficients from the highest power: x² is [1, 0, 0] and x² + 3x − 2 is [1, 3, −2].
Can I multiply?
Yes, choose multiply.
What does the Polynomial Calculator calculate?
Add, subtract or multiply two polynomials, and evaluate the result. It returns Result coefficients, Value at x and Degree.
What formula does the Polynomial Calculator use?
coefficient-wise arithmetic; multiplication is convolution. Coefficients are listed from the highest power down.
What do I need to enter in the polynomial Calculator?
You enter First polynomial coefficients (highest power first), Second polynomial coefficients and Operation. Optional: Evaluate at x (optional).
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Coefficient lists, not expressions.
What does the “Operation” option change?
It sets how the polynomial is worked out. The choices are Add, Subtract and Multiply.
What is an example of a polynomial result?
With first polynomial coefficients (highest power first) 1, 2, second polynomial coefficients 3, -1, operation Add and evaluate at x (optional) 2, the result coefficients is 4, 1.
Related calculators
- Polynomial Long Division Calculator — Quotient and remainder of one polynomial divided by another.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Log Calculator — Logarithm of a number to any base.
Sources
- OpenStax, Prealgebra 2e (Rice University): fractions, decimals, percents, ratios and rounding
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.