Remainder Theorem Calculator
This remainder theorem calculator works out P(r), the value of a polynomial at r, which the remainder theorem tells you is exactly the remainder left when you divide the polynomial by (x - r). Instead of carrying out long or synthetic division, you enter the polynomial's coefficients from the highest power down to the constant term, using 0 for any missing power, and a value for r, and the calculator substitutes r into the polynomial to return the result instantly. The results panel shows P(r), the remainder itself, a clear Yes or No answer to whether (x - r) is a factor of the polynomial, and the value of r you entered for reference. A No answer with a non-zero remainder means r is not a root, while a Yes answer with a remainder of zero confirms r is a root and (x - r) divides the polynomial exactly, which is the factor theorem in action. This is useful for checking possible roots, verifying a factorisation, or avoiding a tedious division when all you need is one value. Make sure you list every coefficient in order, including zeros for missing powers, since leaving one out shifts every remaining coefficient and gives the wrong answer. Results are rounded for display, so treat a very small non-zero remainder as effectively zero only if it is explained by rounding in your decimal inputs.
The formula
The remainder theorem: dividing a polynomial P(x) by x minus r leaves a remainder equal to P(r). The factor theorem follows: x minus r is a factor of P exactly when P(r) = 0.
Worked example
For P(x) = x^2 minus 1 and r = 1, P(1) = 1 minus 1 = 0, so the remainder is 0 and x minus 1 is a factor. Enter 1, 0, -1 and 1 to confirm.
Frequently asked questions
What is the remainder theorem?
Dividing P(x) by x minus r leaves a remainder of P(r), so you only need to substitute r, not do the division.
What is the factor theorem?
The special case where P(r) = 0, which means x minus r divides P exactly and r is a root.
How do I enter the polynomial?
List the coefficients from the highest power down to the constant, using 0 for missing powers.
Who this calculator is for
This calculator is for algebra students and anyone needing a quick, reliable result.
What this calculator assumes
- You enter valid numbers.
- The standard method is applied.
- Results are rounded for display.
Formula and sources
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