Ruffini's Rule — Definition, Formula & Examples

Ruffini's Rule is a shortcut for dividing a polynomial by a linear factor of the form

(x - c)

. It uses only the coefficients of the polynomial and repeated multiplication-and-addition steps to find the quotient and remainder.

Given a polynomial $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ and a divisor $(x - c)$ , Ruffini's Rule is a synthetic algorithm that produces the coefficients $b_{n-1}, b_{n-2}, \ldots, b_0$ of the quotient polynomial $Q(x)$ and the remainder $r$ such that $P(x) = (x - c)\,Q(x) + r$ , by iterating $b_k = a_k + c \cdot b_{k+1}$ from the leading coefficient downward.

How It Works

Write down the value

c

(from the divisor

x - c

) and list all coefficients of the polynomial in descending degree order, inserting 0 for any missing powers. Bring the leading coefficient straight down. Multiply it by

c

, then add the result to the next coefficient. Repeat this multiply-then-add pattern for every remaining coefficient. The final numbers you obtain are the coefficients of the quotient polynomial (one degree lower than the original), and the very last number is the remainder.

Worked Example

Problem: Divide

2x^3 + 3x^2 - 5x + 6

(x - 2)

using Ruffini's Rule.

Setup: The divisor is

(x - 2)

, so

c = 2

. List the coefficients: 2, 3, −5, 6.

Step 1: Bring down the leading coefficient 2. Multiply by

c

2 \times 2 = 4

. Add to the next coefficient:

3 + 4 = 7

Step 2: Multiply 7 by

c

7 \times 2 = 14

. Add to the next coefficient:

-5 + 14 = 9

Step 3: Multiply 9 by

c

9 \times 2 = 18

. Add to the last coefficient:

6 + 18 = 24

Result: The bottom row is 2, 7, 9, 24. The first three numbers are the quotient coefficients; the last is the remainder.

2x^3 + 3x^2 - 5x + 6 = (x - 2)(2x^2 + 7x + 9) + 24

Answer: The quotient is

2x^2 + 7x + 9

with a remainder of

24

Why It Matters

Ruffini's Rule is the standard quick method taught in Algebra 2 and Precalculus for testing potential rational roots of polynomials. When combined with the Rational Root Theorem, it lets you systematically factor higher-degree polynomials without performing full long division each time.

Common Mistakes

Mistake: Using the wrong sign for

c

. When dividing by

(x + 3)

, students use

c = 3

instead of

c = -3

Correction: Always rewrite the divisor in the form

(x - c)

. Since

x + 3 = x - (-3)

, you must use

c = -3

in the algorithm.