Polynomial Modulus Form
Example Data Table
| Dividend coefficients | Modulus coefficients | Coefficient modulus | Quotient | Remainder |
|---|---|---|---|---|
| 1, 0, -1 | 1, -1 | None | x + 1 | 0 |
| 2, 3, 4 | 1, 1 | None | 2x + 1 | 3 |
| 1, 0, 1 | 1, 1 | 5 | x + 4 | 2 |
| 1, 2, 0, 3 | 1, 0, 1 | None | x + 2 | -x + 1 |
Formula Used
Polynomial modulus follows the division identity:
f(x) = q(x)m(x) + r(x)
Here, f(x) is the dividend, m(x) is the modulus polynomial, q(x) is the quotient, and r(x) is the remainder.
The remainder must satisfy:
deg(r(x)) < deg(m(x))
When a coefficient modulus p is supplied, every coefficient is reduced modulo p. The leading-term factor becomes:
factor = lead(r) × inverse(lead(m)) mod p
This keeps all quotient and remainder coefficients inside the chosen coefficient system.
How to Use This Calculator
- Enter dividend coefficients from highest degree to constant term.
- Enter modulus polynomial coefficients in the same order.
- Optionally set a coefficient modulus for finite-field style reduction.
- Add an x value when you also want polynomial evaluations.
- Choose the visible decimal precision for results.
- Press the calculate button to display quotient, remainder, checks, and steps above the form.
- Use the CSV or PDF buttons after calculation to export the current result summary.
Frequently Asked Questions
1. What does polynomial modulus mean?
It means reducing one polynomial by another through long division. The final remainder is the modulus result, and its degree stays lower than the modulus polynomial’s degree.
2. How should I enter coefficients?
Enter them in descending power order, separated by commas. For example, x² − 3x + 2 becomes 1, -3, 2.
3. What happens if I leave coefficient modulus blank?
The calculator performs ordinary polynomial division over real-number coefficients. Quotient and remainder are shown without modular coefficient reduction.
4. Why can coefficient modulus fail?
If the leading coefficient of the modulus polynomial has no modular inverse under your chosen coefficient modulus, the long-division step cannot proceed correctly.
5. Why is the remainder degree important?
A valid polynomial modulus result must always have a smaller degree than the modulus polynomial. That rule ensures the remainder is unique for the chosen system.
6. What does the identity check verify?
It checks whether dividend = quotient × modulus polynomial + remainder. This confirms that the displayed decomposition matches the actual division result.
7. Can I evaluate the result at a specific x?
Yes. Enter an x value and the calculator will evaluate the dividend, modulus polynomial, quotient, and remainder using the same number.
8. When is this useful?
It is useful in algebra courses, coding theory, finite-field arithmetic, symbolic simplification, cryptography practice, and quotient-ring calculations.