# How do you use synthetic division to divide 4x^3 + 7x^2 – 13x – 3 by x + 3?

Jul 30, 2015

$\textcolor{red}{\frac{4 {x}^{3} + 7 {x}^{2} - 13 x - 3}{x + 3} = 4 {x}^{2} - 5 x + 2 - \frac{9}{x + 3}}$

#### Explanation:

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

$| 4 \text{ "color(white)(1)7color(white)(1)-13" " } - 3$
$| \textcolor{w h i t e}{1}$
stackrel("——————————————)

Step 2. Put the divisor at the left.

$\textcolor{red}{- 3} | 4 \text{ "color(white)(1)7color(white)(1)-13" " } - 3$
$\text{ } \textcolor{w h i t e}{1} |$
" "color(white)(1)stackrel("——————————————)

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

$- 3 | 4 \text{ "color(white)(1)7color(white)(1)-13" " } - 3$
$\text{ } \textcolor{w h i t e}{1} |$
" "color(white)(1)stackrel("——————————————)
$\text{ } \textcolor{w h i t e}{1} | \textcolor{red}{4}$

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

$- 3 | 4 \text{ "color(white)(1)7color(white)(1)-13" " } - 3$
$\text{ } \textcolor{w h i t e}{1} | \textcolor{w h i t e}{1} \textcolor{red}{- 12}$
" "color(white)(1)stackrel("——————————————)
$\text{ } \textcolor{w h i t e}{1} | 4$

Step 5. Add down the column.

$- 3 | 4 \text{ "color(white)(1)7color(white)(1)-13" " } - 3$
$\text{ } \textcolor{w h i t e}{1} | - 12$
" "color(white)(1)stackrel("——————————————)
$\text{ } \textcolor{w h i t e}{1} | 4 \textcolor{w h i t e}{1} \textcolor{red}{- 5}$

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

$- 3 | 4 \text{ "color(white)(1)7color(white)(1)-13" " } - 3$
$\text{ "color(white)(1)|-12" "color(white)(1)15" } \textcolor{w h i t e}{1} - 6$
" "stackrel("——————————————)
color(blue)(" "color(white)(1)|4-5" "" "2)" "color(white)(1)color(red)(-9)

$\frac{4 {x}^{3} + 7 {x}^{2} - 13 x - 3}{x + 3} = 4 {x}^{2} - 5 x + 2 - \frac{9}{x + 3}$

Check:

$\left(x + 3\right) \left(4 {x}^{2} - 5 x + 2 - \frac{9}{x + 3}\right) = \left(x + 3\right) \left(4 {x}^{2} - 5 x + 2\right) - 9$

$= 4 {x}^{3} - 5 {x}^{2} + 2 x + 12 {x}^{2} - 15 x + 6 - 9 = 4 {x}^{3} + 7 {x}^{2} - 13 x - 3$