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Remainder Theorem Calculator
Enter your polynomial coefficients above and click "Evaluate f(c)" to see results.The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x − c), the remainder is exactly f(c). This means you can evaluate any polynomial at any point simply by performing synthetic division — the last number in the bottom row equals f(c).
The Factor Theorem is a direct corollary: if f(c) = 0, then (x − c) is a factor of f(x). These two theorems together provide a powerful bridge between evaluation and factoring. Instead of plugging in values manually (which involves large exponents), synthetic division gives the same answer with simpler arithmetic.
For cubic equations, the Remainder Theorem is especially useful for root verification. After finding candidate roots via the Rational Root Theorem, you can quickly confirm which ones are actual roots by checking whether f(c) = 0. This is faster and less error-prone than direct substitution, particularly for large coefficients.
Evaluate f(c) for any value c without computing large powers directly — synthetic division handles it cleanly.
After finding candidate roots, the Remainder Theorem instantly confirms which candidates are actual roots.
The theorem beautifully connects the concepts of division, evaluation, and factoring into one unified framework.
When dividing by (x+3), the evaluation point is c = −3, not c = 3. The theorem uses (x MINUS c).
The Remainder Theorem is not limited to cubics. It works for polynomials of any degree.
The remainder from division equals f(c). Don't confuse the quotient (a polynomial) with the remainder (a number).
| Theorem | f(x) ÷ (x−c) has remainder f(c) |
| Factor Test | f(c) = 0 means (x−c) is a factor |
| Method | Use synthetic division for efficiency |
| Works For | Polynomials of any degree |
| Key Benefit | Avoids computing large powers directly |
Synthetic division gives you the leftover quotient quadratic *and* the remainder. This tool bypasses the quotient and purely gives you the remainder.
Yes! The remainder <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">R</span> is literally the <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">y</span>-coordinate on the graph when <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x = c</span>.
Congratulations! You've found a root of the equation through the Factor Theorem.
The Factor Theorem is a special case of the Remainder Theorem. If the remainder f(c) = 0, then (x - c) is a factor of the polynomial.
Yes, the Remainder Theorem works for polynomials of any degree, not just cubics. It is a universal tool for evaluating polynomial values.
