Remainder theorem calculator
Calculator
The calculator needs JavaScript. The tables below give the same numbers for common cases.
How it is worked out
- The remainder theorem says the remainder of P(x) ÷ (x − r) is P(r).
- Synthetic division gives 5.
- Direct substitution gives 5. They agree.
| Inputs | Answer |
|---|---|
| Coefficients, highest power first: 1 -2 -5 6; Divide by (x − r), r =: 3 | Dividing x³ − 2x² − 5x + 6 by (x − 3) leaves remainder P(3) = 0. |
| Coefficients, highest power first: 2 3 -1 4; Divide by (x − r), r =: -2 | Dividing 2x³ + 3x² − x + 4 by (x + 2) leaves remainder P(-2) = 2. |
How to use it
The remainder theorem says that when a polynomial P(x) is divided by (x − r), the remainder is P(r), the value of the polynomial at x = r. So you can find the remainder by substituting r instead of dividing.
The factor theorem is the special case where the remainder is 0: then (x − r) is a factor of P(x). The calculator shows the remainder by synthetic division and by direct substitution so you can see that they agree.
Frequently asked questions
What is the remainder theorem?
The remainder of P(x) ÷ (x − r) equals P(r).
How do I know if (x − r) is a factor?
If P(r) = 0 the remainder is 0 and (x − r) is a factor.
How do I divide by x + 2?
Use r = −2, because x + 2 = x − (−2).
Method: standard arithmetic shown step by step; no outside data is used. Checked 2026-11-12.
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