Question

How do you simplify and divide `(x^3+3x^2+3x+2)/(x^2+x+1)`?

Answer 1

The remainder is `=0` and the quotient is `=(x+2)`

Explanation:

Let's do a long division

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

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

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

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

`color(white)(aaaaaaaaaa)` `0+0+0`

The remainder is `=0` and the quotient is `=(x+2)`

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

Answer 2

`"The Quotient is "(x+2)" and the Remainder "0`.

Explanation:

Recall that, `(x+1)^3=x^3+3x^2+3x+1,` hence,

`"The Nr.="x^3+3x^2+3x+2=(x^3+3x^2+3x+1)+1`

`=(x+1)^3+1^3,` and,

Using, `a^3+b^3=(a+b)(a^2-ab+b^2)`, we have,

`"The Nr.="{{(x+1)+1)}{(x+1)^2-(x+1)(1)+1^2}`

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

`=(x+2)(x^2+x+1)`.

`"Therefore, the Exp.="{(x+2)(x^2+x+1)}/(x^2+x+1)`

`=(x+2)`.

`"Hence, the Quotient is "(x+2)" and the Remainder "0,` as

derived by Respected Narad T.