Synthetic Division

Synthetic division is a compact numerical shortcut for dividing any polynomial by a linear binomial of the form $x - b$ . Instead of writing out all the polynomial notation, we keep only the coefficients and apply a simple multiply-and-add rule. The result is the quotient and remainder in a fraction of the time.

Quick Reference

Step	Action
Setup	Write $b$ to the left, then the coefficients of the dividend in order (use 0 for missing powers)
Row 1	Bring down the first coefficient unchanged
Repeat	Multiply the last result by $b$ ; write under the next coefficient; add
Result	Bottom row gives coefficients of $q (x)$ (one degree lower), last entry is $r$
For $x + b$	Use $- b$ in place of $b$ (since $x + b = x - (- b)$ )

Deriving the Algorithm

When we divide a general cubic $a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}$ by $x - b$ , the long-division process produces a quotient whose coefficients satisfy a simple pattern. Writing $c_{n - 1}$ for the leading coefficient of $q (x)$ , each successive coefficient is obtained by:

$c_{i} = a_{i + 1} + b \cdot c_{i + 1} .$

In other words: multiply the coefficient just obtained by $b$ , then add the next coefficient of the dividend.

The synthetic division tableau: the bottom row gives the coefficients of <math xmlns= — **Figure 1** The synthetic division tableau: the bottom row gives the coefficients of $q (x)$ followed by the remainder $r$ .

Summary of synthetic division rules:

$c_{n - 1} = a_{n}$ (first coefficient of quotient equals leading coefficient of dividend)
$c_{i} = a_{i + 1} + b \cdot c_{i + 1}$ for $0 \leq i \leq n - 2$ (each subsequent coefficient)
$r = a_{0} + b \cdot c_{0}$ (remainder)

The general synthetic division tableau for a degree- $n$ dividend is:

$\begin{array}{crrrrrr} b & a_{n} & a_{n - 1} & a_{n - 2} & \dots & a_{1} & a_{0} \\ + & c_{n - 1} b & c_{n - 2} b & \dots & c_{1} b & c_{0} b \\ c_{n - 1} & c_{n - 2} & c_{n - 3} & \dots & c_{0} & r \end{array}$

Worked Examples

Example 1.

Divide $3 x^{4} - 5 x^{3} - 4 x^{2} + 3 x - 2$ by $x - 2$ .

Solution

Here

b = 2

and the coefficients of the dividend are $3, -5, -4, 3, -2$.

\begin{array}{crrrrr} 2 & 3 & - 5 & - 4 & 3 & - 2 \\ + & 6 & 2 & - 4 & - 2 \\ 3 & 1 & - 2 & - 1 & - 4 \end{array}

Reading off the bottom row: the quotient coefficients are $3, 1, -2, -1$ (giving a degree-3 quotient) and the remainder is

- 4

q (x) = 3 x^{3} + x^{2} - 2 x - 1, r = - 4.

Therefore:

3 x^{4} - 5 x^{3} - 4 x^{2} + 3 x - 2 = (x - 2) (3 x^{3} + x^{2} - 2 x - 1) + (- 4) .

Important: If the dividend is missing any power of $x$ , insert a $0$ coefficient as a placeholder. For example, $x^{4} - 1 = x^{4} + 0 x^{3} + 0 x^{2} + 0 x - 1$ , so use coefficients $1, 0, 0, 0, -1$.

Example 2.

Divide $x^{4} - 1$ by $x + 1$ .

Solution

Since

x + 1 = x - (- 1)

, we use

b = - 1

. The dividend

x^{4} - 1

has missing intermediate terms, so its coefficients are $1, 0, 0, 0, -1$.

\begin{array}{crrrrr} - 1 & 1 & 0 & 0 & 0 & - 1 \\ + & - 1 & 1 & - 1 & 1 \\ 1 & - 1 & 1 & - 1 & 0 \end{array}

The remainder is $0$, confirming that

x + 1

divides

x^{4} - 1

exactly.

q (x) = x^{3} - x^{2} + x - 1, r = 0.

Dividing by $α x - β$

Synthetic division in its standard form requires a divisor with leading coefficient 1. When the divisor is $α x - β$ , rewrite it as $α (x - β / α)$ , perform synthetic division by $x - β / α$ , then adjust the quotient:

Rewrite the divisor as 
  α
  
  
    (
    x
    −
    
      
        β
        α
      
    
    )
  
.
Divide the polynomial synthetically by 
  x
  −
  
    
      β
      α
    
  
, obtaining 
  q
  (
  x
  )
 and 
  r
.
The quotient corresponding to divisor 
  α
  x
  −
  β
 is 
  
    
      
        q
        (
        x
        )
      
      α
    
  
 and the remainder is 
  r
.

Why this technique works

f (x) = q (x) (x - β / α) + r

, multiply and divide by

α

f (x) = \frac{q (x)}{α} \cdot α (x - \frac{β}{α}) + r = \frac{q (x)}{α} (α x - β) + r .

q (x) / α

is the quotient and

r

is the remainder when dividing by

α x - β

Example 3.

Divide $3 x^{3} - 11 x^{2} + 18 x - 3$ by $3 x - 2$ .

Solution

Write

3 x - 2 = 3 (x - 2 / 3)

and divide synthetically by

x - 2 / 3

(so

b = 2 / 3

\begin{array}{crrrr} \frac{2}{3} & 3 & - 11 & 18 & - 3 \\ + & 2 & - 6 & 8 \\ 3 & - 9 & 12 & 5 \end{array}

The synthetic quotient is

3 x^{2} - 9 x + 12

and the remainder is $5$. Dividing the quotient by

α = 3

q (x) = \frac{3 x^{2} - 9 x + 12}{3} = x^{2} - 3 x + 4.

Therefore:

3 x^{3} - 11 x^{2} + 18 x - 3 = (3 x - 2) (x^{2} - 3 x + 4) + 5.

Frequently Asked Questions

When can I use synthetic division?

Synthetic division requires the divisor to be a linear polynomial (or reducible to one after factoring out a scalar). It works for divisors of the form

x - b

x + b

(use

- b

), or

α x - β

(adjust as described above). For quadratic or higher-degree divisors, use long division or undetermined coefficients.

What if I forget a zero coefficient?

You will get the wrong quotient and remainder. Always check that the dividend has a coefficient for every power from degree

n

down to degree 0 before starting synthetic division. Insert explicit zeros for any missing powers.

What does a remainder of 0 tell me?

A remainder of 0 means that

x - b

divides

f (x)

exactly, i.e.,

x - b

is a factor of

f (x)

. Equivalently,

b

is a root of

f

f (b) = 0

. This is the Factor Theorem.

Quick Reference

Deriving the Algorithm

Worked Examples

Dividing by $α x - β$

Frequently Asked Questions

Quick Links

Social Media

Synthetic Division

Quick Reference

Deriving the Algorithm

Worked Examples

Dividing by α x − β

Frequently Asked Questions

Quick Links

Social Media

Dividing by $α x - β$