Question

What is the quotient `( x ^ { 3} + 3x ^ { 2} + 5x + 3) \div ( x + 1)`?

Answer 1

The quotient is `=x^2+2x+3`

Explanation:

Let's perform a long division

`color(white)(aaaa)` `x+1` `color(white)(aaaa)` `|` `x^3+3x^2+5x+3` `color(white)(aaaa)` `|` `x^2+2x+3`

`color(white)(aaaaaaaaa)` `color(white)(aaaaaa)` `x^3+x^2`

`color(white)(aaaaaaaaa)` `color(white)(aaaaaaa)` `0+2x^2+5x`

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

`color(white)(aaaaaaaaa)` `color(white)(aaaaaaaaaaaa)` `0+3x+3`

`color(white)(aaaaaaaaa)` `color(white)(aaaaaaaaaaaaaa)` `+3x+3`

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

The quotient is `=x^2+2x+3`

Answer 2

`x^2+2x+3`

Explanation:

One way is to use the divisor as a factor in the numerator.

`"consider the numerator"`

`color(red)(x^2)(x+1)color(magenta)(-x^2)+3x^2+5x+3`

`=color(red)(x^2)(x+1)+color(red)(2x)(x+1)color(magenta)(-2x)+5x+3`

`=color(red)(x^2)(x+1)+color(red)(2x)(x+1)+color(red)(3)(x+1)color(magenta)(-3)+3`

`=color(red)(x^2)(x+1)+color(red)(2x)(x+1)+color(red)(3)(x+1)`

`"quotient " =color(red)(x^2+2x+3)`