Calculator
Example data table
| Dividend P(x) coefficients | Divisor D(x) coefficients | Quotient Q(x) | Remainder R(x) | Divisible? |
|---|---|---|---|---|
| 1, -5, 6 | 1, -2 | x - 3 | 0 | Yes |
| 1, 0, -1 | 1, -1 | x + 1 | 0 | Yes |
| 2, 1, 0 | 1, 0, 1 | 2 | 1 | No |
Formula used
- Polynomial long division: Find polynomials Q(x) and R(x) so that P(x)=D(x)·Q(x)+R(x), with deg(R)<deg(D).
- Divisibility rule: D(x) divides P(x) exactly when R(x)=0 (all remainder coefficients are zero within eps).
- Remainder theorem (linear case): For D(x)=x−a, the remainder is P(a). So x−a divides P(x) iff P(a)=0.
- Factor theorem: x−a is a factor of P(x) exactly when a is a root of P(x).
How to use this calculator
- Write each polynomial as a coefficient list from highest power to constant.
- Paste the dividend list into P(variable) and the divisor list into D(variable).
- Set eps to control rounding; smaller is stricter.
- Press Check Divisibility to see quotient and remainder.
- If the remainder is zero, the divisor is a factor of the dividend.
- Use Download CSV or Download PDF after a calculation.
FAQs
1) What does “divisible” mean for polynomials?
A divisor D(x) divides P(x) when P(x)=D(x)·Q(x) with no remainder. This calculator checks if the computed remainder is zero within your tolerance.
2) Why does the calculator ask for coefficients?
Coefficients avoid ambiguity and make long division reliable. Enter values from the highest power down to the constant term, using commas or spaces.
3) What is eps and why might I change it?
eps is the zero-check tolerance. With decimal inputs, tiny rounding errors can appear. Increase eps slightly if you expect clean zeros but see tiny leftovers.
4) Can I test divisibility by x − a quickly?
Yes. For a linear divisor, the remainder equals P(a). If P(a)=0, then x−a is a factor. The result box shows a helpful linear hint.
5) What if the divisor degree is larger than the dividend degree?
Then the quotient is zero and the remainder is the dividend itself. In that case, divisibility only happens when the dividend is the zero polynomial.
6) Do you support fractions like 3/4?
Yes. You can type fractions in the coefficient list (for example, 3/4). They are converted to decimals internally before long division.
7) How are the CSV and PDF exports created?
After you submit, the calculator stores the summary in your session. The export buttons send that stored summary as CSV or a simple PDF report.