Calculator Inputs
Enter coefficients from the highest power to the constant term. Example: 2, -3, 5, -7 represents 2x^3 - 3x^2 + 5x - 7.
Example Data Table
| Polynomial P(x) | Divisor | a value | Remainder P(a) | Quotient |
|---|---|---|---|---|
| 2x3 - 3x2 + 5x - 7 | x - 2 | 2 | 7 | 2x2 + x + 7 |
| x4 - 6x2 + 8 | x + 1 | -1 | 3 | x3 - x2 - 5x + 5 |
| 3x2 + 4x - 9 | x - 0.5 | 0.5 | -6.25 | 3x + 5.5 |
Formula Used
Remainder Theorem: If a polynomial P(x) is divided by x - a, then the remainder is R = P(a).
Horner / Synthetic Division Recurrence: Start with the leading coefficient, multiply by a, add the next coefficient, and repeat until the last value appears. The last value is the remainder, and all earlier values form the quotient coefficients.
- For coefficients cn, cn-1, ..., c0, begin with bn = cn.
- Then compute bk = ck + a · bk+1 moving toward the constant term.
- The final term b0 is the remainder, while the earlier b-values define the quotient polynomial.
How to Use This Calculator
- Enter polynomial coefficients from the highest degree down to the constant term.
- Type the value a from the divisor form x - a. For x + 3, use -3.
- Choose the decimal precision you want for the displayed result.
- Click Calculate Remainder to place the result block above the form.
- Review the remainder, quotient, synthetic table, and theorem check.
- Use the CSV and PDF buttons to save the calculation for classwork, tutoring, or revision notes.
Frequently Asked Questions
1. What does the remainder theorem say?
It says the remainder from dividing a polynomial P(x) by x - a equals P(a). You can find the remainder by substitution instead of long division.
2. Why do I enter coefficients instead of a full expression?
Coefficient input is faster to validate and perfect for synthetic division. It also avoids parsing issues from spaces, missing multiplication signs, or different exponent styles.
3. How do I handle a divisor like x + 4?
Rewrite it as x - (-4). That means your a value is -4. The calculator then evaluates P(-4) and returns the correct remainder.
4. Does the calculator also find the quotient?
Yes. Synthetic division produces both outputs together. The final value is the remainder, and all earlier synthetic results become the quotient coefficients.
5. Can I use decimals or negative numbers?
Yes. The calculator accepts decimal and negative coefficients, as well as decimal or negative values of a, making it useful for advanced exercises.
6. What if the remainder is zero?
A zero remainder means x - a is a factor of the polynomial. In other words, a is a root, and the divisor divides the polynomial exactly.
7. Is this useful for checking homework answers?
Yes. You can compare manual substitution, synthetic division, and the displayed theorem check in one place. That makes errors easier to spot and explain.
8. What input order should I follow?
Always enter coefficients from the highest power down to the constant. If a term is missing, include 0 for that power so the order stays correct.