Question

How do you use synthetic division to divide `(x^4 + 2x^3 - 2x - 1) ÷ (x^3 + 3x^2 + 3x + 1)`?

Answer 1

`x-1`

Explanation:

First, write down the coefficients of each term of the dividend (don't forget `0x^2`):

`color(white)(XX)1color(white)(XX)2color(white)(XX)0color(white)(X)-2color(white)(X)-1`

On the left, write down the negative coefficients of each term in the divisor excluding the first term:

`color(white)(X)-3color(white)(X)-3color(white)(X)-1|color(white)(XX)1color(white)(XX)2color(white)(XX)0color(white)(X)-2color(white)(X)-1color(white)(X)`

The solution will look like this:

`color(white)(X)-3color(white)(X)-3color(white)(X)-1|color(white)(XX)1color(white)(XX)2color(white)(XX)0color(white)(X)-2color(white)(X)-1color(white)(X)`
`color(white)(color(white)(X)-3color(white)(X)-3color(white)(X)-1)|color(white)(XX)color(white)(X)-3color(white)(XX)3color(white)(XX)0color(white)(XXX)0color(white)(X)`
`color(white)(color(white)(X)-3color(white)(X)-3color(white)(X)-1)|color(white)(XX)color(white)(XXX)color(white)(x)-3color(white)(XX)3color(white)(XxX)0color(white)(X)`
`color(white)(color(white)(X)-3color(white)(X)-3color(white)(X)-1)|ul(color(white)(XX)color(white)(XXX)color(white)(XXX)color(white)(X)-1color(white)(XX)1color(white)(X))`
`color(white)(color(white)(X)-3color(white)(X)-3color(white)(X)-1)color(white)(XXX)1color(white)(X)-1color(white)(XX)0color(white)(XX)0color(white)(XXX)0color(white)(X)`

Therefore, the quotient is `x-1`.