What if the divisor is a linear binomial like $(ax - b)$?

You must manipulate the division to yield a leading coefficient of 1. Divide the numerator and the denominator by $a$. The new divisor is $(x - b/a)$, so $c = b/a$. Perform synthetic division using the modified dividend coefficients. The resulting quotient and remainder will be correct for the original problem.

What does a remainder of 0 signify?

By the Factor Theorem, a remainder of 0 means that $(x - c)$ is a precise factor of the polynomial $P(x)$, and $c$ is a root of the equation $P(x) = 0$.

Can synthetic division be used for non-linear divisors?

Standard synthetic division only works for linear divisors. For divisors of degree 2 or higher (e.g., $x^2 + 3$), you must use polynomial long division.

How to divide polynomials using synthetic division

Synthetic division is a shorthand algorithmic method for dividing a polynomial by a linear binomial of the form $(x - c)$ . It compresses the standard long division process by operating exclusively on the coefficients of the polynomial. This method applies only when the divisor has a degree of 1 and a leading coefficient of 1. If the divisor has a leading coefficient other than 1, algebraic manipulation is required before proceeding.

The setup

Identify the constant $c$ in the divisor $(x - c)$ . Write $c$ to the left of the division bracket. Extract the coefficients of the dividend polynomial in descending order of degree. You must insert a $0$ for any missing power of the variable. Write these coefficients in a row next to $c$ . Draw a horizontal line below the coefficients, leaving space for a second row of numbers.

The steps

Bring the leading coefficient of the dividend straight down below the horizontal line. 2. Multiply this bottom value by $c$ and write the product in the next column, above the line. 3. Add the numbers in this column vertically and write the sum below the line. 4. Repeat the multiplication and addition process until all columns are filled. The final number below the line is the remainder. The preceding numbers are the coefficients of the quotient polynomial, starting exactly one degree lower than the original dividend.

Checking the result

Verify the computation using the division algorithm: $P(x) = D(x)Q(x) + R$ . Multiply your derived quotient $Q(x)$ by the divisor $D(x)$ , then add the remainder $R$ . Expand and simplify the expression. The result must exactly match the original dividend $P(x)$ .

Common errors

The most frequent error is omitting zero placeholders for missing terms in the dividend, which misaligns the entire degree structure. A second common error is using the wrong sign for $c$ . If the divisor is $(x + 4)$ , the value of $c$ is $-4$ , not $4$ .

Worked example

Divide $P(x) = 2x^3 - 3x^2 + 4x - 5$ by $(x - 2)$ .

The divisor is $(x - 2)$ , so $c = 2$ . The coefficients of $P(x)$ are $2, -3, 4, -5$ . Setup: $2 \mid 2 \quad -3 \quad 4 \quad -5$ . Step 1: Bring down the $2$ . Step 2: Multiply $2 imes 2 = 4$ . Write $4$ under $-3$ . Add: $-3 + 4 = 1$ . Step 3: Multiply $1 imes 2 = 2$ . Write $2$ under $4$ . Add: $4 + 2 = 6$ . Step 4: Multiply $6 imes 2 = 12$ . Write $12$ under $-5$ . Add: $-5 + 12 = 7$ . The bottom row is $2 \quad 1 \quad 6 \quad \mid \quad 7$ . The remainder is $7$ . The quotient coefficients are $2, 1, 6$ . Because the dividend is cubic, the quotient is quadratic. Quotient: $2x^2 + x + 6$ . Final answer: $2x^2 + x + 6 + \frac{7}{x - 2}$ .

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References: OpenStax College Algebra, Section 5.4: Dividing Polynomials · Larson Precalculus, Chapter 2: Polynomial and Rational Functions