Synthetic division is the fast, compact way to divide a polynomial by a linear factor (x − c) — the same operation full polynomial long division performs, but stripped down to just the coefficients. It's the standard classroom tool for testing whether a value is a root, factoring higher-degree polynomials, and evaluating a polynomial at a point without expanding it.
How the bring-down table works
Write the polynomial's coefficients across the top, in descending order of degree, making sure to include a 0 placeholder for any power that doesn't appear. Bring the first coefficient straight down — that's the start of the running sum. Then, for every column that follows, multiply the current running sum by c and add the result to that column's coefficient. The number you get becomes both the next entry in the running-sum row and the input to the next multiplication.
Every running sum except the very last one becomes a coefficient of the quotient polynomial, which has one degree less than the original. The very last running sum is the remainder.
Why it also gives you P(c) for free
The Remainder Theorem states that dividing a polynomial P(x) by (x − c) always leaves a remainder equal to P(c) — the value of the polynomial at x = c. That's exactly what the bring-down/multiply/add process computes: it's algebraically identical to Horner's method for evaluating a polynomial. So synthetic division isn't just a division shortcut — it's simultaneously the fastest way to evaluate P(c) without substituting c into every term and simplifying by hand.
Using it to test factors and find roots
Because the remainder equals P(c), a remainder of 0 tells you two things at once: (x − c) divides the polynomial exactly, and c is a root of the polynomial. This makes synthetic division the standard tool for testing candidate roots (often integer factors of the constant term, per the Rational Root Theorem) one at a time until a factor is found, then repeating the process on the resulting lower-degree quotient to factor completely.
Limits: this calculator's synthetic division only handles a divisor of the exact form (x − c) — a monic linear factor. Dividing by something like (2x − 3) requires either factoring out the leading coefficient first or falling back to full polynomial long division.