Question

How do you find the quotient of `(8c^3+6c-5)div(4c-2)` using long division?

Answer 1

The remainder is `=-1` and the quotient is `=2c^2+c+2`

Explanation:

Let's perform the long division

`4c-2` `color(white)(aaaa)` `|` `8c^3+0c^2+6c-5` `color(white)(aaaa)` `|` `2c^2+c+2`

`color(white)(aaaaaaaaaaa)` `8c^3-4c^2`

`color(white)(aaaaaaaaaaaa)` `0+4c^2+6c`

`color(white)(aaaaaaaaaaaaaa)` `+4c^2-2c`

`color(white)(aaaaaaaaaaaaaaa)` `+0+8c-5`

`color(white)(aaaaaaaaaaaaaaaaaaa)` `+8c-4`

`color(white)(aaaaaaaaaaaaaaaaaaaa)` `+0-1`

Therefore,

`(8c^3+0c^2+6c-5)/(4c-2)=2c^2+c+2-1/(4c-2)`

Answer 2

`(8c^3 +6c-5) div (4c-2) = 2c^2 +c +2 " rem "1`

or:

`(8c^3 +6c-5) div (4c-2) = 2c^2 +c +2 + (-1)/(4c-2)`

Explanation:

• In each case, `4c` is divided into the term with the highest power of `c` available.
• the quotient is multiplied by both terms at the side.
• change signs to subtract the expressions. (shown in red)

`color(white)(wwwwwwww)2c^2+c+2`
`4c-2 )bar(8c^3 " "+6c-5)larr" "` desc order, no term in `c^2`
`color(white)(www)ul(color(red)(-)8c^3color(red)(+)4c^2)" "larr` subtract
`color(white)(wwwwwwwww)4c^2+6c`
`color(white)(wwn.....ww)color(red)(-)ul(4c^2color(red)(+)2c)`
`color(white)(wwwwwwwwwww)+8c-5`
`color(white)(wwwwwwwwwww)color(red)(-)ul( 8c color(red)(+)4)`
`color(white)(wwwwwwwwwwwwww)-1" "` remainder

`(8c^3 +6c-5) div (4c-2) = 2c^2 +c +2 " rem "1`

or:

`(8c^3 +6c-5) div (4c-2) = 2c^2 +c +2 + (-1)/(4c-2)`