Question

How do you use polynomial synthetic division to divide `(2x^3+x^2+2x+1)div(x+1/2)` and write the polynomial in the form `p(x)=d(x)q(x)+r(x)`?

Answer 1

The answer is `=(2x^2+2)(x+1/2)`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `-1/2` `color(white)(aaaa)` `|` `color(white)(aa)` `2` `color(white)(aaaaaaa)` `1` `color(white)(aaaaaa)` `2` `color(white)(aaaaaaa)` `1`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaa)` `-1` `color(white)(aaaaaa)` `0` `color(white)(aaaaa)` `-1`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaa)` `2` `color(white)(aaaaaaa)` `0` `color(white)(aaaaaa)` `2` `color(white)(aaaaaa)` `color(red)(0)`

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

Therefore,

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