Enter your expression
Use expanded single-variable form like 4x^3 - 2x + 7.
Formula used
A polynomial in one variable follows the form P(x) = anxn + an-1xn-1 + ... + a1x + a0.
Each exponent must be a nonnegative integer. Coefficients may be integers or decimals. The variable cannot appear in a denominator, radical, function name, or fractional exponent.
Degree: the highest exponent with a nonzero coefficient.
Evaluation: P(c) = Σ akck.
Derivative: P'(x) = Σ k·akxk-1.
How to use this calculator
1. Enter an expanded expression in one variable.
2. Set the variable symbol, usually x.
3. Add an optional evaluation value.
4. Choose display decimals for cleaner output.
5. Press Determine Polynomial.
6. Review the yes or no result, degree, standard form, derivative, and coefficient graph.
7. Export the result as CSV or PDF when needed.
Example data table
| Input expression | Polynomial? | Reason | Degree or note |
|---|---|---|---|
| 3x^4 - 2x + 7 | Yes | All exponents are whole and nonnegative. | 4 |
| 0.5x^2 - 1.75x + 3 | Yes | Decimal coefficients are allowed. | 2 |
| x^-2 + 1 | No | Negative exponents are not polynomial exponents. | Invalid |
| 1/x + 2 | No | The variable appears in a denominator. | Invalid |
| sqrt(x) + 5 | No | Radicals create fractional exponents. | Invalid |
| (x + 1)^2 | Needs expansion first | The expression is polynomial after expansion. | Enter x^2 + 2x + 1 |
FAQs
1. What makes an expression a polynomial?
A polynomial has terms with constant coefficients and nonnegative integer exponents. The chosen variable cannot appear inside denominators, radicals, logarithms, or trigonometric functions.
2. Are decimal coefficients allowed?
Yes. Values like 0.25x^3 - 1.8x + 9 are still polynomials because decimals affect coefficients, not the exponent rules.
3. Why is 1/x not a polynomial?
Because 1/x equals x^-1. That creates a negative exponent, which breaks the polynomial definition.
4. Does x^(1/2) count as a polynomial term?
No. A fractional exponent represents a radical-style term. Polynomial exponents must be whole numbers starting from zero.
5. Can this tool check more than one variable?
This version checks one chosen variable at a time. It is designed for single-variable expressions in expanded form.
6. Why should I expand parentheses first?
The checker reads standard expanded terms directly. Expansion avoids ambiguity and lets the calculator measure degree, coefficients, and graphable term data cleanly.
7. What does the degree tell me?
The degree is the highest exponent with a nonzero coefficient. It strongly affects graph shape, turning behavior, and end behavior.
8. Why is the zero polynomial special?
Every coefficient equals zero, so no highest nonzero exponent exists. Many textbooks call its degree undefined, though some software uses zero for convenience.