# How do you divide (18x^4+9x^3+3x^2)/(3x^2+1) using synthetic division?

Apr 3, 2018

$\frac{18 {x}^{4} + 9 {x}^{3} + 3 {x}^{2}}{3 {x}^{2} + 1} = 6 {x}^{2} + 3 x - 1 + \frac{- 3 x + 1}{3 {x}^{2} + 1}$

#### Explanation:

Synthetic division is more commonly used with linear divisors and especially with monic linear divisors. It is possible to use a form of it with divisors of degree $> 1$ as in this example, but it may be easier to use in the form of long division of coefficients...

color(white)(30000010")")underline(color(white)(001)6color(white)(000)3color(white)(00)-1color(white)(00-000-1)
$3 \textcolor{w h i t e}{00} 0 \textcolor{w h i t e}{00} 1 \textcolor{w h i t e}{0} \text{)} \textcolor{w h i t e}{00} 18 \textcolor{w h i t e}{000} 9 \textcolor{w h i t e}{00 -} 3 \textcolor{w h i t e}{00 -} 0 \textcolor{w h i t e}{00 -} 0$
$\textcolor{w h i t e}{30000010 \text{)} 00} \underline{18 \textcolor{w h i t e}{000} 0 \textcolor{w h i t e}{00 -} 6}$
$\textcolor{w h i t e}{30000010 \text{)} 0000000} 9 \textcolor{w h i t e}{00} - 3 \textcolor{w h i t e}{00 -} 0$
$\textcolor{w h i t e}{30000010 \text{)} 0000000} \underline{9 \textcolor{w h i t e}{00 -} 0 \textcolor{w h i t e}{00 -} 3}$
$\textcolor{w h i t e}{30000010 \text{)} 0000000900} - 3 \textcolor{w h i t e}{00} - 3 \textcolor{w h i t e}{00 -} 0$
$\textcolor{w h i t e}{30000010 \text{)} 000000090} \underline{\textcolor{w h i t e}{0} - 3 \textcolor{w h i t e}{00 -} 0 \textcolor{w h i t e}{00} - 1}$
$\textcolor{w h i t e}{30000010 \text{)} 0000000900 - 300} - 3 \textcolor{w h i t e}{00 -} 1$

The first important thing to note is the inclusion of $0$'s for missing powers of $x$ in both the dividend $18 {x}^{4} + 9 {x}^{3} + 3 {x}^{2}$ represented as $18 \textcolor{w h i t e}{0} 9 \textcolor{w h i t e}{0} 3 \textcolor{w h i t e}{0} 0 \textcolor{w h i t e}{0} 0$ and the divisor $3 {x}^{2} + 1$ represented as $3 \textcolor{w h i t e}{0} 0 \textcolor{w h i t e}{0} 1$.

The actual process of long division is similar to long division of numbers.

In our example, we arrive at a quotient $6 \textcolor{w h i t e}{0} 3 \textcolor{w h i t e}{0} - 1$ representing $6 {x}^{2} + 3 x - 1$ and remainder $- 3 \textcolor{w h i t e}{0} 1$ representing $- 3 x + 1$.