Substitute values into polynomials with classroom-ready calculations. Track powers, coefficients, derivatives, and Horner evaluation instantly. Save neat outputs for homework, revision, reports, and practice.
General polynomial: \( P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)
Substitution: Replace \( x \) with the chosen number \( k \), so \( P(k) = a_nk^n + a_{n-1}k^{n-1} + ... + a_1k + a_0 \).
Horner form: \( (((a_nk + a_{n-1})k + a_{n-2})k + ... )k + a_0 \)
First derivative: \( P'(x) = na_nx^{n-1} + ... + 2a_2x + a_1 \)
Second derivative: \( P''(x) = n(n-1)a_nx^{n-2} + ... + 2a_2 \)
| Polynomial | Substitution value | Result | First derivative value |
|---|---|---|---|
| 3x³ - 2x² + 5x - 7 | x = 2 | 19 | 33 |
| 2x⁴ + x² - 9 | x = -1.5 | 3.375 | -24 |
| -x³ + 4x + 6 | x = 3 | -9 | -23 |
It means replacing the variable in a polynomial with a chosen number, then simplifying every term and adding the results.
List them from highest degree to constant term. For 4x² - 3x + 8, enter 4, -3, 8.
Yes. The calculator accepts positive, negative, and decimal numbers, then evaluates each power and term accordingly.
Horner form reduces repeated power calculations, checks the same answer efficiently, and is useful for manual verification and programming.
They show how fast the polynomial changes at the substituted value, plus how its curvature behaves through the second derivative.
Yes. Include zero coefficients for missing powers, such as 5, 0, -2 for 5x² - 2.
The table lists each power, coefficient, substituted power value, and final term contribution before the polynomial total is summed.
Yes. Use the built-in CSV button for spreadsheet use or the PDF button for a clean printable report.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.