Calculator
Enter coefficients from highest degree to constant. Use commas, spaces, or new lines.
Example data table
Use these sample inputs to verify your setup.
| Coefficients | x | Method | P(x) | P'(x) | P''(x) | ∫ from 0 to x |
|---|---|---|---|---|---|---|
| 2, -5, 3, -7 | 4 | Horner | 53 | 59 | 38 | 17.333333 |
| 1, 0, -9 | 3 | Direct | 0 | 6 | 2 | -18 |
Formula used
- Polynomial: P(x) = anxn + an−1xn−1 + … + a1x + a0
- Direct evaluation: P(x) = Σ akxk
- Horner’s method: P(x) = (…((anx + an−1)x + an−2)… )x + a0
- Derivative: P'(x) = Σ k·akxk−1
- Second derivative: P''(x) = (P')'(x)
- Definite integral (optional): ∫ab P(t) dt, computed using power rules.
How to use this calculator
- Write coefficients from highest power down to the constant.
- Enter the x value where you want P(x) evaluated.
- Pick Horner for speed, or Direct for term inspection.
- Set decimal places for rounding and display precision.
- Optionally add limits a and b for a definite integral.
- Press Evaluate polynomial to see results above.
- Use the CSV and PDF buttons to export your work.
FAQs
1) What order should I enter coefficients?
Enter coefficients from the highest degree down to the constant term. For example, “2, -5, 3, -7” means 2x³ − 5x² + 3x − 7.
2) What is Horner’s method, and why use it?
Horner’s method rewrites the polynomial into nested multiplication. It reduces multiplications and can be more numerically stable than computing large powers directly.
3) Can I use decimals or scientific notation?
Yes. Coefficients and x can be decimals like 0.125 or scientific notation like 1e-6. Separate values using commas, spaces, or new lines.
4) Why do I see both Direct and Horner tables?
The selected method controls the main P(x) output, but both tables help you verify work. Direct shows term contributions; Horner shows the step-by-step recurrence.
5) What do P'(x) and P''(x) represent?
P'(x) is the slope (rate of change) of the polynomial at x. P''(x) measures curvature, indicating how the slope itself is changing at that point.
6) How is the optional definite integral computed?
If you enter both limits a and b, the page applies the power rule term-by-term and evaluates the antiderivative at b minus the value at a.
7) What happens if some coefficients are zero?
Zero coefficients are allowed. They simply mean that term is missing in the polynomial. The string formatter may omit zero terms for readability.
8) Why might my results differ from a calculator app?
Differences usually come from rounding. Increase decimal places to compare more precisely. Very large degrees or huge numbers can also amplify floating-point effects.