# How do you use polynomial synthetic division to divide (3x^3-x+4)div(x-2/3) and write the polynomial in the form p(x)=d(x)q(x)+r(x)?

Oct 8, 2017

$3 {x}^{3} - x + 4 = \left(x - \frac{2}{3}\right) \left(3 {x}^{2} + 2 x + \frac{1}{3}\right) + 4 \frac{2}{9}$

#### Explanation:

To divide $3 {x}^{3} + 0 {x}^{2} - x + 4$ by $x - \frac{2}{3}$

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

$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} 3 \text{ "color(white)(X)0color(white)(XX)-1" "" } 4$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Two Put $\frac{2}{3}$ in the divisor at the left as $x - \frac{2}{3} = 0$ gives $x = \frac{2}{3}$

$\frac{2}{3} | \textcolor{w h i t e}{X} 3 \text{ "color(white)(X)0color(white)(XX)-1" "" } 4$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Three Drop the first coefficient of the dividend below the division symbol.

$\frac{2}{3} | \textcolor{w h i t e}{X} 3 \text{ "color(white)(X)0color(white)(XX)-1" "" } 4$
$\textcolor{w h i t e}{X} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{X} | \textcolor{w h i t e}{X} \textcolor{b l u e}{3}$

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

$\frac{2}{3} | \textcolor{w h i t e}{\times} 3 \text{ "color(white)(X)0color(white)(XX)-1" "" } 4$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X X x} 2$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{3}$

$\frac{2}{3} | \textcolor{w h i t e}{X} 3 \text{ "color(white)(X)0color(white)(XX)-1" "" } 4$
$\textcolor{w h i t e}{X} | \textcolor{w h i t e}{X X X X} 2$
color(white)(1)stackrel("—————————————)
$\textcolor{w h i t e}{X} | \textcolor{w h i t e}{X} \textcolor{b l u e}{3} \textcolor{w h i t e}{X X x} \textcolor{red}{2}$

Six Repeat Steps Four and Five until you can go no farther.

$\frac{2}{3} | \textcolor{w h i t e}{X} 3 \text{ "color(white)(X)0color(white)(XX)-1" "" } 4$
$\textcolor{w h i t e}{X} | \text{ } \textcolor{w h i t e}{X x X} 2 \textcolor{w h i t e}{X X X} \frac{4}{3} \textcolor{w h i t e}{X x X} \frac{2}{9}$
color(white)(1)stackrel("—————————————)
$\textcolor{w h i t e}{X} | \textcolor{w h i t e}{X} \textcolor{b l u e}{3} \textcolor{w h i t e}{X x X} \textcolor{red}{2} \textcolor{w h i t e}{X X X} \textcolor{red}{\frac{1}{3}} \textcolor{w h i t e}{X x X} \textcolor{red}{4 \frac{2}{9}}$

Hence remainder is $4 \frac{2}{9}$ and quotient is $3 {x}^{2} + 2 x - \frac{1}{3}$

and hence $3 {x}^{3} - x + 4 = \left(x - \frac{2}{3}\right) \left(3 {x}^{2} + 2 x + \frac{1}{3}\right) + 4 \frac{2}{9}$