Polynomial Calculator
Add, subtract, and multiply polynomials with ease. Enter polynomials in standard form (3x² + 2x - 1) or as comma-separated coefficients and get instant results with degree, leading coefficient, and constant term.
Polynomial Calculator
How to Use the Polynomial Calculator
Enter two polynomials using either standard notation (like 3x^2 + 2x - 1) or comma-separated coefficients (like -1, 2, 3 for the same polynomial). Select the operation you want to perform and click Calculate.
Input Formats
The calculator accepts polynomials in two formats:
- Standard Form: Use x for the variable and ^ for exponents (e.g., 3x^2 + 2x - 1)
- Coefficient Form: List coefficients from constant term to highest power, separated by commas (e.g., -1, 2, 3)
Polynomial Operations
Addition
To add polynomials, combine like terms (terms with the same power of x). For example: (3x² + 2x - 1) + (x² - 4x + 5) = 4x² - 2x + 4
Subtraction
To subtract polynomials, distribute the negative sign and combine like terms. For example: (3x² + 2x - 1) - (x² - 4x + 5) = 2x² + 6x - 6
Multiplication
To multiply polynomials, use the distributive property (FOIL for binomials). Each term in the first polynomial multiplies each term in the second, then combine like terms.
Key Polynomial Terms
- Degree: The highest power of x with a non-zero coefficient
- Leading Coefficient: The coefficient of the highest-degree term
- Constant Term: The term with no x (power of 0)
- Like Terms: Terms with the same power of x
Worked Example: Multiplying Two Binomials
Multiply (2x + 3) by (x − 4). Distribute each term of the first polynomial across the second: 2x·x = 2x², 2x·(−4) = −8x, 3·x = 3x, and 3·(−4) = −12. Combining the like terms −8x + 3x gives −5x, so the product is 2x² − 5x − 12. Note that the degree of a product always equals the sum of the degrees of the factors (1 + 1 = 2 here) — a quick sanity check the calculator's degree readout makes easy.
Common Applications
Polynomial arithmetic is fundamental in:
- Algebra and precalculus courses
- Engineering and physics calculations
- Computer graphics and animation
- Signal processing and control systems
- Cryptography and coding theory
Frequently Asked Questions
How do I enter a polynomial into the calculator?+
You can type standard notation such as 3x^2 + 2x - 1, using ^ for exponents, or enter comma-separated coefficients from the constant up to the highest power (-1, 2, 3 represents the same polynomial). Both inputs produce identical results, so use whichever matches your source material.
What operations does this polynomial calculator perform?+
It adds, subtracts, and multiplies two polynomials, automatically combining like terms and sorting the result in standard descending-power form. Alongside the result it reports the degree, leading coefficient, and constant term of the answer.
What happens to the degree when I add or multiply polynomials?+
When multiplying, degrees add: a degree-2 times a degree-3 polynomial always yields degree 5. When adding or subtracting, the result's degree is at most the larger of the two degrees, but it can drop if leading terms cancel — for example (x² + x) − (x² − 3) = x + 3 falls from degree 2 to degree 1.
Can it handle negative and fractional coefficients?+
Yes. Terms like -4x^3 or 0.5x are parsed correctly in standard form, and decimals are accepted in coefficient form as well. The output keeps exact arithmetic on the coefficients you supply, so results match hand calculation.
Is the polynomial calculator free and mobile-friendly?+
Yes. It is free, needs no account, and all parsing and arithmetic run locally in your browser, so it works instantly and privately. The interface is responsive and usable on phones, which makes it practical for checking homework anywhere.
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