# How do you use polynomial synthetic division to divide (3x^2-2x+1) div(x-1) and write the polynomial in the form p(x)=d(x)q(x)+r(x)?

Mar 23, 2017

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

#### Explanation:

To divide $\left(3 {x}^{2} - 2 x + 1\right)$ by $\left(x - 1\right)$ using synthetic division, we should take following steps

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

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

Two Put the divisor at the left, but for this put $x - 1 = 0$, which gives $x = 1$

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

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

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

Four Multiply the result of $1 \times \textcolor{b l u e}{3} = 3$, here $1$ comes from divisor, and put the product in the next column.

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

$1 | \textcolor{w h i t e}{X} 3 \text{ } \textcolor{w h i t e}{X} - 2 \textcolor{w h i t e}{X X X X} 1$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X X X X} 3$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} \textcolor{b l u e}{3} \textcolor{w h i t e}{X X X x} \textcolor{b l u e}{1}$

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

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

WE find that quotient is $\left(x + 1\right)$ and remainder is $2$.

Hence $\left(3 {x}^{2} - 2 x + 1\right) = \textcolor{b l u e}{\left(3 x + 1\right)} \left(x - 1\right) + \textcolor{red}{2}$