Question

How do you long divide `(5x^3+x^2-x+3) / (x+1)`?

Answer 1

The answer is `=5x^2-4x+3`

Explanation:

Let's do the long division
`color(white)(aaaa)` `5x^3+x^2-x+3` `color(white)(aaa)` `∣` `x+1`
`color(white)(aaaa)` `5x^3+5x^2` `color(white)(aaaaaaaaa)` `∣` `5x^2-4x+3`
`color(white)(aaaaaa)` `0-4x^2-x`
`color(white)(aaaaaaaa)` `-4x^2-4x`
`color(white)(aaaaaaaaaaaa)` `0+3x+3`
`color(white)(aaaaaaaaaaaaaaaa)` `3x+3`
`color(white)(aaaaaaaaaaaaaaaaa)` `0+0`

So `(5x^3+x^2-x+3)` is divisible by `(x+1)`
You can test this by doing
let `f(x)=5x^3+x^2-x+3`
the `f(-1)=-5+1+1+3=0`
This is the remainder theorem