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Remainder Theorem Calculator

Find the remainder of a polynomial division using the remainder theorem.

Introduction

Remainder Theorem Calculator — Find the remainder of a polynomial division using the remainder theorem. Enter Coefficients (highest degree first, comma separated), r to get an instant, accurate result.

Formula

By the remainder theorem, the remainder of polynomial p(x) divided by (x − r) equals p(r), computed directly via synthetic evaluation of the coefficients at r.

Step-by-step

  1. Enter the Coefficients (highest degree first, comma separated).
  2. Enter the r.
  3. Click Calculate to see your result instantly.

Real-world example

Example: With Coefficients (highest degree first, comma separated) = 1, -3, 2, r = 2, the Remainder Theorem Calculator gives Remainder: 0.

Frequently Asked Questions

How is this different from long division?
The remainder theorem lets you find just the remainder by evaluating the polynomial at r, without performing the full division.
What does a remainder of 0 mean?
It means (x − r) is a factor of the polynomial, i.e. r is a root.
What inputs does the Remainder Theorem Calculator need?
You'll need to provide: Coefficients (highest degree first, comma separated), r. All fields use sensible defaults, so you can see a working example immediately and then adjust the values to match your own numbers.
How accurate is the Remainder Theorem Calculator?
The Remainder Theorem Calculator applies the formula exactly as calculated — By the remainder theorem, the remainder of polynomial p(x) divided by (x − r) equals p(r), computed directly via synthetic evaluation of the coefficients at r. — so results are precise for the inputs you provide. Accuracy depends on entering correct, realistic input values.
Is the Remainder Theorem Calculator free to use?
Yes, the Remainder Theorem Calculator is completely free, requires no signup, and runs instantly in your browser.

About the Remainder Theorem Calculator

The Remainder Theorem Calculator uses a real, verifiable formula — By the remainder theorem, the remainder of polynomial p(x) divided by (x − r) equals p(r), computed directly via synthetic evaluation of the coefficients at r. — so results are accurate every time, not an approximation.