The formula
P(x) ÷ (x − r) = Q(x) + R, with R = P(r)
Why it works
Synthetic division — Ruffini's rule — is long division stripped to the coefficients. Bring down the first one, multiply by r, add to the next, and repeat. What you get is the quotient's coefficients and, at the end, the remainder.
How to solve it by hand
- Write the coefficients in order, skipping none — use 0 for missing powers
- Bring down the first coefficient unchanged
- Multiply it by r and add the result to the next coefficient
- Repeat to the end: the last number is the remainder
What is worth knowing
The remainder theorem is what makes this more than a shortcut: dividing P(x) by (x − r) always leaves P(r) as the remainder, so a remainder of zero proves r is a root. That gives you a fast root test and, when it works, a factorisation — the polynomial becomes (x − r) × quotient, and the quotient is one degree lower. Repeating the process is the standard way to factor a cubic or quartic by hand. The one thing that trips people up is missing powers: x³ + 1 has coefficients 1, 0, 0, 1, and forgetting the zeros gives nonsense.