Question

How do you long divide `(x^3+3x^2+x)div (x+2)`?

Answer 1

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

Explanation:

You can either do a long division or use the remainder theorem

Let's do the long division

`color(white)(aaaa)` `x^3+3x^2+x` `color(white)(aaaa)` `∣` `x+2`

`color(white)(aaaa)` `x^3+2x^2` `color(white)(aaaaaaaa)` `∣` `x^2+x-1`

`color(white)(aaaaaa)` `0+x^2+x`

`color(white)(aaaaaaaa)` `+x^2+2x`

`color(white)(aaaaaaaaa)` `+0-x`

`color(white)(aaaaaaaaaaaaa)` `-x`

`color(white)(aaaaaaaaaaaaa)` `-x-2`

`color(white)(aaaaaaaaaaaaaaa)` `0+2`

So

`(x^3+3x^2+x)/(x+2)=x^2+x-1+2/(x+2)`

Let's try the remainder theorem

Let `f(x)=x^3+3x^2+x`

Then. `f(-2)=(-2)^3+3*(-2)^2-2`

`=-8+12-2=2`

The remainder is `=2`