Synthetic Division of Polynomials

What is synthetic division?

Synthetic division is a shortcut for dividing a polynomial by a linear factor $(x - c)$. It's faster than long division because you only work with the coefficients.

Step-by-step: Divide $x^3 - 4x^2 + x + 6$ by $(x - 2)$

Step 1 — Set up. Write the coefficients of the dividend in a row: $[1, -4, 1, 6]$. Write $c = 2$ to the left (from $x - 2$, we use $+2$).

$2 \;\big|\; 1 \quad -4 \quad 1 \quad 6$

Step 2 — Bring down the first coefficient: $1$.

$2 \;\big|\; 1 \quad -4 \quad 1 \quad 6$ $\quad\quad\quad \underline{\downarrow}$ $\quad\quad\quad 1$

Step 3 — Multiply and add for each column:

Column 2: $1 \times 2 = 2$. Add to $-4$: $-4 + 2 = -2$.

Column 3: $-2 \times 2 = -4$. Add to $1$: $1 + (-4) = -3$.

Column 4: $-3 \times 2 = -6$. Add to $6$: $6 + (-6) = 0$.

Bottom row: $[1, -2, -3, 0]$.

Step 4 — Read the answer. The last number is the remainder (= 0). The other numbers are the quotient coefficients: $1x^2 - 2x - 3$.

$x^3 - 4x^2 + x + 6 = (x - 2)(x^2 - 2x - 3) + 0$

Step 5 — Since remainder = 0, $(x - 2)$ is a factor! We can factor further: $x^2 - 2x - 3 = (x - 3)(x + 1)$.

Complete factorization: $x^3 - 4x^2 + x + 6 = (x - 2)(x - 3)(x + 1)$.

The carry-multiply-add pattern

At each step: multiply the bottom number by $c$, write it under the next coefficient, then add down. Repeat.

When to use synthetic division

• Only when dividing by $(x - c)$ (a linear factor with leading coefficient 1).
• For other divisors (like $x^2 + 2$ or $2x - 3$), use polynomial long division instead.

Try it in the visualization

Adjust $c$ and the coefficients. The carry-multiply-add process is animated column by column. If the remainder is 0, the visualization confirms $(x - c)$ is a factor.