# How do you evaluate \frac{z ^ { 3} + 0z ^ { 2} - z + 42}{z - 7}?

Aug 12, 2018

$\left({z}^{3} + 0 {z}^{2} - z + 42\right) = \left(z - 7\right) \left({z}^{2} + 7 z + 48\right) + \left(378\right)$

#### Explanation:

$\left({z}^{3} + 0 {z}^{2} - z + 42\right) \div \left(z - 7\right)$

Using synthetic division :

We have , $p \left(z\right) = \left({z}^{3} + 0 {z}^{2} - z + 42\right) \mathmr{and} \text{divisor : } z = 7$

We take , coefficients of $p \left(z\right) \to 1 , 0 , - 1 , 42$

. $7 |$ $1 \textcolor{w h i t e}{\ldots \ldots . .} 0 \textcolor{w h i t e}{\ldots \ldots} - 1 \textcolor{w h i t e}{\ldots \ldots . .} 42$
$\underline{\textcolor{w h i t e}{\ldots}} |$ ul(0color(white)( ........)7color(white)(........)49color(white)(.......)336
color(white)(......)1color(white)(.......)7color(white)(........)48color(white)(.......)color(violet)(ul|378|
We can see that , quotient polynomial :

$q \left(z\right) = {z}^{2} + 7 z + 48 \mathmr{and} \text{the Remainder} = 378$

Hence ,

$\left({z}^{3} + 0 {z}^{2} - z + 42\right) = \left(z - 7\right) \left({z}^{2} + 7 z + 48\right) + \left(378\right)$