# How do you use synthetic division to divide 3x^3+4x^2-7x+1 by 3x-2?

Jul 30, 2015

$\textcolor{red}{\frac{3 {x}^{3} + 4 {x}^{2} - 7 x + 6}{3 x - 2} = {x}^{2} + 2 x - 1 - \frac{1}{3 x - 2}}$

#### Explanation:

We use a slightly modified table when the coefficient of $x$ does not equal $1$. Note the extra lines.

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

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

Step 2. Put the divisor at the left.

$\text{ "" "|3" "4" "-7" " " " } 1$
$\text{ } \textcolor{red}{2} \textcolor{w h i t e}{1} |$
" "stackrel("——————————————)

Step 3. Write the coefficient of $x$ below the division line

$\text{ "" "|3" "4" "-7" " " " } 1$
$\text{ "2|" "color(white)(1)2 " "" "4" } - 2$
" "stackrel("——————————————)
$\text{ } \textcolor{w h i t e}{1} |$
$\textcolor{red}{/ 3} \textcolor{w h i t e}{1} |$

Step 4. Drop the first coefficient of the dividend below the division symbol.

$\text{ "" "|3" "4" "-7" " " " } 1$
$\text{ "2|" "color(white)(1)2 " "" "4" } - 2$
" "stackrel("——————————————)
$\text{ } \textcolor{w h i t e}{1} | \textcolor{red}{3}$
$/ 3 \textcolor{w h i t e}{1} |$

Step 5. Divide the dropped value by the coefficient of $x$ and place the result in the row below.

$\text{ "" "|3" "4" "-7" " " " } 1$
$\text{ "2|" "color(white)(1)2 " "" "4" } - 2$
" "stackrel("——————————————)
$\text{ "" } | 3$
$/ 3 \textcolor{w h i t e}{1} | \textcolor{red}{1}$

Step 6. Multiply the result by the constant, and put the product in the next column.

$\text{ "" "|3" "4" "-7" " " " } 1$
$\text{ "2|" } \textcolor{w h i t e}{1} \textcolor{red}{2}$
" "stackrel("——————————————)
$\text{ "" } | 3$
$/ 3 \textcolor{w h i t e}{1} | 1$

Step 7. Add down the column.

$\text{ "" "|3" "4" "-7" " " " } 1$
$\text{ "2|" } \textcolor{w h i t e}{1} 2$
" "stackrel("——————————————)
$\text{ "" "|3" } \textcolor{red}{6}$
$/ 3 \textcolor{w h i t e}{1} | 1$

Step 8. Repeat Steps 5, 6, and 7 until you can go no farther.

$\text{ "" "|3" "4" "-7" " " " } 1$
$\text{ "2|" "color(white)(1)2 " "" "4" } - 2$
" "stackrel("——————————————)
$\text{ "" "|3" "6" "-3" } \textcolor{red}{- 1}$
$/ 3 \textcolor{w h i t e}{1} | 1 \text{ "2" } - 1$

$\frac{3 {x}^{3} + 4 {x}^{2} - 7 x + 6}{3 x - 2} = {x}^{2} + 2 x - 1 - \frac{1}{3 x - 2}$

Check:

$\left(3 x - 2\right) \left({x}^{2} + 2 x - 1 - \frac{1}{3 x - 2}\right) = \left(3 x - 2\right) \left({x}^{2} + 2 x - 1\right) - 1$

$= 3 {x}^{3} + 6 {x}^{2} - 3 x - 2 {x}^{2} - 4 x + 2 - 1 = 3 {x}^{3} + 4 {x}^{2} - 7 x + 1$

Jul 30, 2015

$\left(3 {x}^{3} + 4 {x}^{2} - 7 x + 1\right) \div \left(3 x - 2\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$= {x}^{2} + 2 x - 1$$\textcolor{w h i t e}{\text{XXXX}}$Remainder: -1

#### Explanation:

Note that this is simply an alternate approach to Ernest Z's answer. Some people may find one approach easier to understand than the other.

Set up as standard long division:

$3 x$ " goes into " $3 {x}^{3}$$\textcolor{w h i t e}{\text{XXXX}}$$\rightarrow {x}^{2}$ times:

Multiply $3 x - 2$ by ${x}^{2}$ and write the product below the line:

Subtract:

"Bring down" the $- 7 x$

$3 x$ " goes into " $6 {x}^{2}$$\textcolor{w h i t e}{\text{XXXX}}$$\rightarrow 2 x$ times
...and so on...

The Remainder of $\left(- 1\right)$ may be simply noted as a remainder or written as a fraction (-1/(3x-2))_