Remainder Theorem Evaluator

The Remainder Theorem states that evaluating $P(a)$ gives the exact remainder $R$ when $P(x)$ is divided by $(x - a)$.

Coefficients Value of 'a' (divisor $x - a$)

Remainder Verification

Compare substitution directly against synthetic division (Ruffini).

Factor Theorem Checker

A polynomial $P(x)$ has a factor $(x - a)$ if and only if $P(a) = 0$.

Coefficients Candidate root 'a' (for binomial $x - a$)

Factorization Conclusion

Is $(x - a)$ a factor of $P(x)$?

Rational Roots Calculator (p/q)

Find all possible divisors $p$ (constant term) and $q$ (leading coefficient).

Coefficients

Candidates & Verification

Click any $\pm(p/q)$ candidate to check if $P(r) = 0$.

Polynomial Builder

Define desired roots and construct the expanded polynomial.

Roots separated by spaces (e.g.: -2 1 3) Leading Coefficient ($a_n$)

Expanded Polynomial

Simplified result ready to solve or practice.

Ruffini Quiz Challenge

Solve timed questions and test your accuracy.

Current Question

Practice the table

Pick a difficulty or set your own polynomial and divisor $r$, then fill in the Ruffini table.

Difficulty Coefficients Divisor $r$ (for $x - r$) Quick divisors

Your score

Correct verifications build your streak.