Synthetic Division

How Synthetic Division Works

Synthetic division is a streamlined shortcut for dividing a polynomial p(x) by a linear factor (x − k). It replaces long division with a compact table of additions and multiplications.

Setup: Write k in the left cell, then list all coefficients of p(x) across the top row — including a 0 for every missing degree. For example, x³ − 8 = x³ + 0x² + 0x − 8, so you write 1, 0, 0, −8.

Process:

Bring down the first coefficient as-is.
Multiply that value by k; write the product under the next coefficient.
Add the column; write the sum below the line.
Repeat multiply → add until all columns are done.

The bottom row gives the quotient coefficients (one degree lower than p(x)) and the final number is the remainder.

Synthetic division only works when the divisor is exactly (x − k) — a monic linear polynomial. For divisors like (2x − 3) or (x² + 1), use polynomial long division instead.

Remainder & Factor Theorems

Two powerful theorems connect synthetic division to zeros and factors:

Remainder Theorem: When p(x) is divided by (x − k), the remainder equals p(k). In other words, you can evaluate a polynomial at any value k by doing synthetic division — the last number in the bottom row is p(k).
Factor Theorem: (x − k) is a factor of p(x) if and only if p(k) = 0. So if the remainder is zero after synthetic division, you've confirmed a zero of p(x) and found a factor.

Use the Rational Root Theorem to generate candidate values of k, then use synthetic division to test each one quickly. A remainder of 0 means you found a root — and the quotient row gives you a reduced polynomial to factor next.

Polynomial division still tripping you up?