When you divide by $x - r$, long division repeats the same moves with lots of writing that never changes. Synthetic division keeps only the numbers.
Key idea
To divide by $x - r$: write $r$ in the box and the dividend's coefficients in a row (0 for missing powers). Bring down the first coefficient. Then repeat: multiply by $r$, write the result under the next coefficient, and add. The last number is the remainder; the others are the quotient's coefficients, one degree lower.
Why it works
In long division by $x - r$, each round multiplies the divisor by the new quotient term and subtracts. The only number that matters in that subtraction is $-r$ times the quotient coefficient. Subtracting $-r$ times something is the same as adding $r$ times it, so synthetic division uses $r$ and adds instead of subtracting. Every column of the shortcut matches one column of long division.
Worked examples
Worked example: The basic routine
Divide $(x^3 - 4x^2 + x + 6) \div (x - 3)$.
- Box: $3$. Coefficients: $1, -4, 1, 6$.
- Bring down 1. $3 \cdot 1 = 3$; $-4 + 3 = -1$.
- $3 \cdot (-1) = -3$; $1 + (-3) = -2$.
- $3 \cdot (-2) = -6$; $6 + (-6) = 0$.
Answer: Quotient $x^2 - x - 2$, remainder $0$ (so $x - 3$ is a factor).
Worked example: Dividing by $x + 2$
Divide $(2x^3 + 3x^2 - 4) \div (x + 2)$.
- $x + 2 = x - (-2)$, so the box holds $-2$. Coefficients: $2, 3, 0, -4$.
- Bring down 2. $-2 \cdot 2 = -4$; $3 - 4 = -1$.
- $-2 \cdot (-1) = 2$; $0 + 2 = 2$. Then $-2 \cdot 2 = -4$; $-4 - 4 = -8$.
Answer: Quotient $2x^2 - x + 2$, remainder $-8$
Common mistake
Using $-r$ instead of $r$. To divide by $x - 3$ the box holds $3$; to divide by $x + 2$ it holds $-2$. Ask what value of $x$ makes the divisor 0. A second check: the remainder must equal $p(r)$, so for the last example $p(-2) = -16 + 12 - 4 = -8$.