Synthetic Division

What is Synthetic Division?

Synthetic division is a quick way to divide a polynomial by a simple expression like $x - c$ . It uses only the coefficients of the polynomial and skips the full long division process.

When to Use Synthetic Division

You can use synthetic division when the divisor is in the form $x - c$ , where $c$ is a constant (like $x - 2$ or $x + 3$ ). It does not work for divisors with higher powers or coefficients on $x$ (e.g., $2x - 3$ ).

Steps of Synthetic Division

Step 1: Write the Coefficients

Take all the coefficients of the polynomial and write them in a row. Make sure the polynomial is in standard form (highest powers first) and include $0$ for any missing terms.

Example: Divide $2x^3 - 6x^2 + 4x - 8$ by $x - 2$ .

The coefficients are $2, -6, 4, -8$ .

Since the divisor is $x - 2$ , the value of $c$ is $2$ .

Set it up like this:

$\begin{array}{r|rrrr} 2 & 2 & -6 & 4 & -8 \end{array}$

Step 2: Bring Down the First Coefficient:

Write the first coefficient below the line. This starts the process.

$\begin{array}{r|rrrr}2 & 2 & -6 & 4 & -8 \\ & & & & \\ \hline & 2 & & & \\ \end{array}$

Step 3: Multiply and Add:

Multiply the number below the line by $c$ (the divisor value) and write the result under the next coefficient.
Add the column values and write the result below the line.

Step-by-Step Example:

Multiply $2$ (below the line) by $2$ (divisor value): $2 \cdot 2 = 4$ . Write $4$ under $-6$ .
Add $-6 + 4 = -2$ . Write $-2$ below the line.

$\begin{array}{r|rrrr}2 & 2 & -6 & 4 & -8 \\ & & 4 & & \\ \hline & 2 & -2 & & \\ \end{array}$

Multiply $-2$ (below the line) by $2$ (divisor value): $-2 \cdot 2 = -4$ . Write $-4$ under $4$ .

Add $4 + (-4) = 0$ . Write $0$ below the line.

Repeat for the last column:

Multiply $0$ by $2$ : $0 \cdot 2 = 0$ . Write $0$ under $-8$ .
Add $-8 + 0 = -8$ .

$\begin{array}{r|rrrr}2 & 2 & -6 & 4 & -8 \\ & & 4 & -4 & 0 \\ \hline & 2 & -2 & 0 & -8 \end{array}$

Step 4: Interpret the Result:

The numbers below the line represent the coefficients of the quotient.
The last number is the remainder.

Final Answer: The quotient is $2x^2 - 2x + 0$ , or simply $2x^2 - 2x$ . The remainder is $-8$ . So, the result is:

$\frac{2x^3 - 6x^2 + 4x - 8}{x - 2} = 2x^2 - 2x - \frac{8}{x - 2}.$