Question

How do you divide `(x^3+x+3)/(x-5)`?

Answer 1

The remainder is `color(red)(133)` and the quotient is `=x^2+5x+26`

Explanation:

Let's perform a synthetic division

`color(white)(aa)` `5` `color(white)(aaaaa)` `|` `color(white)(aaa)` `1` `color(white)(aaaaa)` `0` `color(white)(aaaaaa)` `1` `color(white)(aaaaaaaaaa)` `3`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaa)` `5` `color(white)(aaaaaa)` `25` `color(white)(aaaaaaaa)` `130`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaa)` `|` `color(white)(aaa)` `1` `color(white)(aaaaa)` `5` `color(white)(aaaaaa)` `26` `color(white)(aaaaaaaa)` `color(red)(133)`

The remainder is `color(red)(133)` and the quotient is `=x^2+5x+26`

`(x^3+x+3)/(x-5)=x^2+5x+26+133/(x-5)`

Answer 2

`x^2+5x+26+133/(x-5)`

Explanation:

`"one way is to use the divisor as a factor in the numerator"`

`"consider the numerator"`

`color(red)(x^2)(x-5)color(magenta)(+5x^2)+x+3`

`=color(red)(x^2)(x-5)color(red)(+5x)(x-5)color(magenta)(+25x)+x+3`

`=color(red)(x^2)(x-5)color(red)(+5x)(x-5)color(red)(+26)(x-5)color(magenta)(+130)+3`

`=color(red)(x^2)(x-5)color(red)(+5x)(x-5)color(red)(+26)(x-5)+133`

`"quotient "=color(red)(x^2+5x+26)," remainder" =133`

`rArr(x^3+x+3)/(x-5)=x^2+5x+26+133/(x-5)`

Answer 3

Quotient`=x^2+5x+26` and remainder`133/(x-5)`

Explanation:

`color(white)(..........)color(white)(.)x^2+5x+26`
`x-5|overline(x^3+0+x+3)`
`color(white)(............)ul(x^3-5x^2)`
`color(white)(......................)5x^2+x`
`color(white)(......................)ul(5x^2-25x)`
`color(white)(..................................)26x+3`
`color(white)(..................................)ul(26x-130)`
`color(white)(................................................)133`

`(x^3+x+3) / (x-5) = x^2+5x+26+133/(x-5)`