# How do you use synthetic division to divide ( 6x^3 - x^2 - 7x + 5 ) / ( x - 3 )?

Sep 20, 2015

$\frac{6 {x}^{3} - {x}^{2} - 7 x + 5}{x - 3} = \textcolor{b l u e}{6 {x}^{2} + 17 x + 44} + \textcolor{red}{\frac{137}{x - 3}}$

#### Explanation:

Step 1. Write only the coefficients of $x$ in the dividend inside an upside-down division symbol.

$\textcolor{w h i t e}{1} | 6 \text{ "-1color(white)(X)-7" "" } 5$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Step 2. Put the divisor at the left.

$3 | 6 \text{ "-1color(white)(X)-7" "" } 5$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Step 3. Drop the first coefficient of the dividend below the division symbol.

$3 | 6 \text{ "-1color(white)(X)-7" "" } 5$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | \textcolor{red}{6}$

Step 4. Multiply the result by the constant, and put the product in the next column.

$3 | 6 \text{ "-1color(white)(X)-7" "" } 5$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X 1} 18$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | \textcolor{b l u e}{6}$

Step 5. Add down the column.

$3 | 6 \text{ "-1color(white)(X)-7" "" } 5$
$\textcolor{w h i t e}{1} | \text{ "color(white)(X1)18" } \textcolor{w h i t e}{1} 51 \textcolor{w h i t e}{X 1} 132$
" "stackrel("—————————————)
color(white)(1)|color(blue)6" "color(red)(color(white)(X)17

Step 6. Repeat Steps 4 and 5 until you can go no farther.

$3 | 6 \text{ "-1color(white)(X)-7" "" } 5$
$\textcolor{w h i t e}{1} | \text{ "color(white)(X1)18" } \textcolor{w h i t e}{1} 51 \textcolor{w h i t e}{X 1} 132$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | \textcolor{b l u e}{6} \text{ "color(white)(X)17" "color(white)(1)44" } \textcolor{w h i t e}{1} \textcolor{red}{137}$

$\frac{6 {x}^{3} - {x}^{2} - 7 x + 5}{x - 3} = 6 {x}^{2} + 17 x + 44 + \frac{137}{x - 3}$
$\left(x - 3\right) \left(6 {x}^{2} + 17 x + 44 + \frac{137}{x - 3}\right) = \left(x - 3\right) \left(6 {x}^{2} + 17 x + 44\right) + 137 = 6 {x}^{3} + 17 {x}^{2} + 44 x - 18 {x}^{2} - 51 x - 132 + 137 = 6 {x}^{3} - {x}^{2} - 7 x + 5$