Remainder Theorem Calculator

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Calculate the remainder of a polynomial division quickly without long division.

Input Parameters

Polynomial coefficients (x³, x², x, const)

Coefficient a (x³)

Coefficient b (x²)

Coefficient c (x)

Constant d

Divide by (x - k)

Calculated Remainder

4.00

Remainder (R)

Evaluated value f(k)

Factor Theorem CheckNot a Factor

Internal Calculation:

f(2) = 0(2)³ + 2(2)² + -5(2) + 6 = 4.00

About this calculator

Quick Reference

Factor TheoremR = 0

Divisorx - k

OperationSubstitution

What is the Remainder Theorem?

The Polynomial Remainder Theorem states that the remainder of the division of a polynomial f(x) by a linear divisor (x - k) is equal to f(k). This is extremely useful because it allows you to find the remainder by simple substitution rather than performing long division or synthetic division.

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Pro Tips

Substitution: If you divide by (x - 3), k = 3. If you divide by (x + 3), k = -3.
Factor Theorem: If the remainder f(k) = 0, then (x - k) is a factor of the polynomial.
Speed: This theorem is the fastest way to check roots of a polynomial.

Fun Facts

"The Remainder Theorem is a key component of the 'Synthetic Division' shortcut."
"It works for any polynomial, regardless of the degree."

Formula: Remainder = f(k)