Calculator Inputs
Use a single stacked page layout. Inside the calculator itself, fields switch to three columns on large screens, two on medium screens, and one on mobile screens.
Example Data Table
| Polynomial Coefficients | Polynomial Form | Divisor | Equivalent Root | Expected Remainder |
|---|---|---|---|---|
| 1, -6, 11, -6 | x³ - 6x² + 11x - 6 | x - 1 | 1 | 0 |
| 2, 3, -5 | 2x² + 3x - 5 | x + 2 | -2 | -3 |
| 4, 0, -9, 7 | 4x³ - 9x + 7 | 2x - 4 | 2 | 21 |
| 3, -1, 8 | 3x² - x + 8 | x - 3 | 3 | 32 |
Formula Used
Standard remainder theorem: If a polynomial P(x) is divided by x - c, the remainder equals P(c).
General linear divisor: For ax + b, first convert it to a(x - r) where r = -b/a. The remainder is then P(r).
Synthetic division recurrence: Bring down the first coefficient, multiply by r, add to the next coefficient, and repeat until the last value appears as the remainder.
How to Use This Calculator
- Enter polynomial coefficients from highest degree to constant term.
- Enter the linear divisor in the form ax + b using separate fields.
- Choose display precision and optional graph limits if needed.
- Press the calculate button to show the result above the form.
- Review the remainder, quotient, synthetic steps, and plotted evaluation point.
- Use the export buttons to save the result as CSV or PDF.
Frequently Asked Questions
1. What does the remainder theorem say?
It states that the remainder from dividing a polynomial P(x) by x - c equals the value of P(c). This turns a division problem into a substitution problem.
2. Can this calculator handle divisors like 2x - 4?
Yes. It converts ax + b into the equivalent theorem value r = -b/a, then evaluates the polynomial at r to get the same remainder.
3. In what order should I enter coefficients?
Always start with the highest degree coefficient and move downward. For 3x² - 5x + 7, enter 3, -5, 7.
4. Why is the graph useful here?
The graph shows the polynomial’s shape and highlights the exact x-value used for the theorem. It helps you connect algebraic evaluation with the plotted point.
5. What does a zero remainder mean?
A zero remainder means the divisor corresponds to a root of the polynomial. In factor language, the related linear term is an exact factor.
6. Does the tool show quotient information too?
Yes. It shows the quotient for x - r from synthetic division and also scales it to match the original divisor ax + b.
7. Can I use decimals or negative values?
Yes. Decimal coefficients, negative coefficients, and decimal divisor terms are supported, provided the divisor’s x coefficient is not zero.
8. What is the fastest way to verify the answer manually?
Find r = -b/a from the divisor, then evaluate the polynomial at r using substitution or Horner’s method. The final value is the remainder.