# How do you use synthetic division to divide x^3- 6x^2+ 11x - 6  by x-1?

Synthetic division is generally used, in finding zeroes (or roots) of polynomials.

#### Explanation:

Well since the polynomial ${x}^{3} - 6 {x}^{2} + 11 x - 6 = \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)$

Thus the quotient of division with $x - 1$ is $\left(x - 2\right) \left(x - 3\right)$ and the remainder is $0$

Sep 11, 2015

See the explanation section.

#### Explanation:

Divide ${x}^{3} - 6 {x}^{2} + 11 x - 6$ by $x - 1$.

First, you let the coefficients of each degree to be used in the division ($1 , - 6 , 11 , - 6$).

Then, dividing by $x - 1$ implies that you use $1$ in your upper left. So, put $1$, then a line (I've used a double line), and then write $\text{ 1" " -6" " 11" " -6 }$ to the right.

$\text{1 } | |$ $\text{ 1" " -6" " 11" " -6 }$
$+$
$\text{ " " }$$- - - - -$

First, bring the first $1$ down to the bottom, and multiply it by the $1$. Put that $1$ below the $- 6$.

$\text{1 } | |$ $\text{ 1" " -6" " 11" " -6 }$
$+$ $\text{ "" " " 1}$
$\text{ " " }$$- - - - -$
$\text{ "" " " 1}$

Then add $- 6 + 1 = - 5$ Put that under the $1$

$\text{1 } | |$ $\text{ 1" " -6 " " 11" " -6 }$
$+$ $\text{ "" " " 1}$
$\text{ " " }$$- - - - -$
$\text{ " " " " 1" " -5}$

Multiply $1 \times - 5 = - 5$ and put the $- 5$ under the $11$. Then add:

$\text{1 } | |$ $\text{ 1" " -6 " " 11 " " -6 }$
$+$ $\text{ "" " " 1 " " -5}$
$\text{ " " }$$- - - - - - - -$
$\text{ " " " " 1 " " -5 " " 6}$

Now $1 \times 6 = 6$, so we put $6$ under $- 6$ and add::

$\text{1 } | |$ $\text{ 1" " -6 " " 11 " " -6 }$
$+$ $\text{ "" "" 1 " " -5 " " 6}$
$\text{ " " }$$- - - - - - - -$
$\text{ "" " " 1 " " -5 " " 6 " |"0 }$

The bottom row ignoring the last number gives us the coefficients of the quotient.
The last number on the bottom row is the remainder (and it is also $P \left(1\right)$).

So the division gives us:

$\frac{{x}^{3} - 6 {x}^{2} + 11 x - 6}{x - 1} = {x}^{2} - 5 x + 6$

You can check the answer by multiplyng:

$\left(x - 1\right) \left({x}^{2} - 5 x + 6\right)$ to make sure we get ${x}^{3} - 6 {x}^{2} + 11 x - 6$.

(I've used Synthetic Division Formatting by Truong-Son R.)