Synthetic Division

Synthetic Division

A shortcut for polynomial long division that can be used when dividing by an expression of the form x – c or x + c. Note: This allows an improper rational expression to be written as a sum of a polynomial and a proper rational expression.

Synthetic division example: 5x³−8x²+9x+12 divided by x−3, showing tableau with result 5x²+7x+30 remainder 102.

See also

Synthetic substitution

Key Formula

\frac{a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0}{x - c} = q_{n-1} x^{n-1} + \cdots + q_1 x + q_0 + \frac{r}{x - c}

Where:

$a_n, a_{n-1}, \ldots, a_0$ = Coefficients of the dividend polynomial, from highest degree to constant term
$c$ = The value from the divisor x − c (use the opposite sign of what appears in the divisor)
$q_{n-1}, \ldots, q_0$ = Coefficients of the quotient polynomial, one degree lower than the dividend
$r$ = The remainder after division

Worked Example

Problem: Divide 2x³ + 3x² − 5x + 6 by x − 2 using synthetic division.

Step 1: Identify c from the divisor x − 2. Here c = 2. Write the coefficients of the dividend in order: 2, 3, −5, 6.

c = 2 \qquad \text{Coefficients: } 2 \;\; 3 \;\; {-5} \;\; 6

Step 2: Bring down the first coefficient (2) unchanged. Multiply it by c and write the result under the next coefficient.

\text{Bring down } 2. \quad 2 \times 2 = 4. \quad \text{Write 4 under the 3.}

Step 3: Add the column: 3 + 4 = 7. Multiply 7 by c = 2 to get 14. Write 14 under −5.

3 + 4 = 7, \quad 7 \times 2 = 14

Step 4: Add the column: −5 + 14 = 9. Multiply 9 by c = 2 to get 18. Write 18 under 6.

{-5} + 14 = 9, \quad 9 \times 2 = 18

Step 5: Add the last column: 6 + 18 = 24. This is the remainder. The bottom row reads 2, 7, 9, 24, giving a quotient of 2x² + 7x + 9 with remainder 24.

\frac{2x^3 + 3x^2 - 5x + 6}{x - 2} = 2x^2 + 7x + 9 + \frac{24}{x - 2}

Answer: 2x² + 7x + 9 with a remainder of 24, or equivalently 2x² + 7x + 9 + 24/(x − 2).

Another Example

This example shows two common variations: dividing by x + c (requiring a negative value of c) and handling missing terms by inserting 0 as a placeholder coefficient.

Problem: Divide x⁴ − 6x² + 8 by x + 2 using synthetic division.

Step 1: The divisor is x + 2 = x − (−2), so c = −2. The dividend is x⁴ + 0x³ − 6x² + 0x + 8. Notice you must include 0 as a placeholder for any missing degree term.

c = -2 \qquad \text{Coefficients: } 1 \;\; 0 \;\; {-6} \;\; 0 \;\; 8

Step 2: Bring down 1. Multiply: 1 × (−2) = −2. Add to next coefficient: 0 + (−2) = −2.

1 \;\;\xrightarrow{\times(-2)}\;\; -2 \quad \Rightarrow \quad 0 + (-2) = -2

Step 3: Multiply: −2 × (−2) = 4. Add: −6 + 4 = −2.

-2 \times (-2) = 4, \quad -6 + 4 = -2

Step 4: Multiply: −2 × (−2) = 4. Add: 0 + 4 = 4.

-2 \times (-2) = 4, \quad 0 + 4 = 4

Step 5: Multiply: 4 × (−2) = −8. Add: 8 + (−8) = 0. The remainder is 0, meaning (x + 2) divides evenly.

4 \times (-2) = -8, \quad 8 + (-8) = 0

Answer: x³ − 2x² − 2x + 4 with remainder 0. So x⁴ − 6x² + 8 = (x + 2)(x³ − 2x² − 2x + 4).

Frequently Asked Questions

When can you use synthetic division instead of polynomial long division?

You can use synthetic division only when the divisor is a linear expression of the form x − c (or x + c). If the divisor has a higher degree, such as x² + 3, or a leading coefficient other than 1, such as 2x − 5, you must use polynomial long division or modify the technique. Some textbooks extend synthetic division to divisors like 2x − 5 by factoring out the 2 first, but the standard method requires the divisor to be monic and linear.

What is the connection between synthetic division and the Remainder Theorem?

The Remainder Theorem states that when you divide a polynomial f(x) by x − c, the remainder equals f(c). Synthetic division naturally computes f(c) as its last entry. This is why the same procedure used for synthetic division is also called synthetic substitution — you are simultaneously dividing and evaluating the polynomial at x = c.

What do you do with missing terms in synthetic division?

If the dividend polynomial is missing a term (for example, x⁴ − 6x² + 8 has no x³ or x term), you must insert 0 as the coefficient for each missing degree. The coefficient list must include every power from the highest degree down to the constant, with no gaps. Forgetting a placeholder is one of the most common errors in synthetic division.

Synthetic Division vs. Polynomial Long Division

	Synthetic Division	Polynomial Long Division
What it divides by	Linear divisors of the form x − c only	Any polynomial divisor of any degree
What you write	Only the coefficients; no variable terms	Full polynomial expressions at each step
Speed	Faster — fewer steps and less writing	Slower — requires multiply, subtract, bring-down at each stage
Risk of error	Lower for linear divisors; sign errors with x + c are common	Higher due to more written algebra, but works in all cases
Result	Quotient coefficients and remainder	Quotient polynomial and remainder polynomial

Why It Matters

Synthetic division appears constantly in Algebra 2 and Precalculus when factoring polynomials, finding zeros, and applying the Rational Root Theorem — you test candidate roots quickly using synthetic division and check whether the remainder is zero. It is also the fastest way to rewrite an improper rational expression as a polynomial plus a proper rational expression, which is useful in partial fraction decomposition and calculus integration.

Common Mistakes

Mistake: Using the wrong sign for c when dividing by x + c.

Correction: Always rewrite the divisor in the form x − c before identifying c. For example, x + 3 means x − (−3), so c = −3, not +3. Using +3 will give completely wrong results.

Mistake: Forgetting to include 0 as a placeholder for missing powers in the dividend.

Correction: List coefficients for every degree from the highest down to the constant. If x⁴ − 6x² + 8 is your dividend, the coefficient list must be 1, 0, −6, 0, 8 — not 1, −6, 8. Skipping the zeros shifts all subsequent computations and produces an incorrect quotient.

Related Terms

Polynomial Long Division — General method that synthetic division shortcuts
Synthetic Substitution — Same procedure used to evaluate f(c)
Improper Rational Expression — Synthetic division rewrites these as mixed form
Proper Rational Expression — The fractional part of the result
Polynomial — The type of expression being divided
Expression — General term for the dividend and divisor
Sum — Result is a sum of polynomial and remainder term