Question

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

Answer 1

`(z^3+0z^2-z+42)=(z-7)(z^2+7z+48 )+(378)`

Explanation:

`(z^3+0z^2-z+42)div(z-7)`

Using synthetic division :

We have , `p(z)=(z^3+0z^2-z+42) and "divisor : " z=7`

We take , coefficients of `p(z) to 1,0 ,-1,42`

. `7|` `1color(white)(........)0color(white)(......)-1color(white)(........)42`
`ulcolor(white)(...)|` `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(z)=z^2+7z+48 and"the Remainder"=378`

Hence ,

`(z^3+0z^2-z+42)=(z-7)(z^2+7z+48 )+(378)`