Question

How do you use the remainder theorem to determine the remainder when the polynomial `2x^4-2x^3+5x` is divided by `x+2`?

Answer 1

The remainder is `=38`

Explanation:

We perform a synthetic division

`color(white)(aaaa)` `-2` `color(white)(aaaa)` `|` `color(white)(aaaa)` `2` `color(white)(aaaa)` `-2` `color(white)(aaaaa)` `0` `color(white)(aaaaaa)` `5` `color(white)(aaaaaa)` `0`

`color(white)(aaaaaa)` `color(white)(aaaaa)` `|` `color(white)(aaa)` `color(white)(aaaaaa)` `-4` `color(white)(aaaa)` `12` `color(white)(aaa)` `-24` `color(white)(aaaaaa)` `38`

`color(white)(aaaaaaaaaa)`------------------------------------------------------------

`color(white)(aaaa)` `color(white)(aaaaaa)` `color(white)(aaaaaa)` `2` `color(white)(aaaaa)` `-6` `color(white)(aaaa)` `12` `color(white)(aaa)` `-19` `color(white)(aaaaaa)` `color(red)(38)`

The remainder is `=38`

The quotient is `=2x^3-6x^2+12x-19`

Verification

Using the remainder theorem

`f(-2)=32+16-10=38`