Question

How do you use the remainder theorem to determine the remainder when the polynomial `(x^3+2x^2-3x+9)div(x+3)`?

Answer 1

The remainder is `color(red)(9)`

Explanation:

The remainder theorem states :

When we divide a polynomial `f(x)` by `(x-c)`, we get

`f(x)=(x-c)q(x)+r(x)`

and

`f(c)=0*q(x)+r=r`

We apply this theorem

`f(x)=(x^3+2x^2-3x+9`

Therefore,

`f(-3)=((-3)^3+2(-3)^2-3(-3)+9`

`=-27+18+9+9=9`

The remainder is `=9`

Let's perform the synthetic division to confirm the results

`color(white)(aa)` `-3` `color(white)(aaaaa)` `|` `color(white)(aaa)` `1` `color(white)(aaaaa)` `2` `color(white)(aaaaaa)` `-3` `color(white)(aaaaa)` `9`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaa)` `-3` `color(white)(aaaaaaaa)` `3` `color(white)(aaaaa)` `0`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaaaa)` `|` `color(white)(aaa)` `1` `color(white)(aaaa)` `-1` `color(white)(aaaaaaa)` `0` `color(white)(aaaaa)` `color(red)(9)`

The remainder is `color(red)(9)` and the quotient is `=x^2-x`

`(x^3+2x^2-3x+9)/(x+3)=x^2-x+9/(x+3)`

Answer 2

The remainder is `9`. See explanation.

Explanation:

According to the Remainder Threorem the remainder of division `P(x)-:(x-a)` is equal to `P(a)`.

Here `a=-3`, so the remainder is:

`P(3)=(-3)^3+2*(-3)^2-3*(-3)+9=`

`=-27+18+9+9=9`