Question

How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems `x^5 + 2x^4 - 3x + 3` divided by x - 1?

Answer 1

The remainder is `=3`

Explanation:

The remainder theorem states that when a polynomial `f(x)` is divided by `x-c`

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

`f(c)=0+r`

Here,

`f(x)=x^5+2x^4-3x+3`

and `(x-1)`

`f(1)=1+2-3+3=3`

The remainder is `=3`

We now perform the synthetic division

`color(white)(aaaa)` `1` `color(white)(aaaa)` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `2` `color(white)(aaaa)` `0` `color(white)(aaaa)` `0` `color(white)(aaaa)` `-3` `color(white)(aaaa)` `3`

`color(white)(aaaa)` ###color(white)(aaaaa)#`|` `color(white)(aaaaa)` ###color(white)(aaaa)#`1` `color(white)(aaaa)` `3` `color(white)(aaaa)` `3` `color(white)(aaaa)` `3` `color(white)(aaaaa)` `0`

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

`color(white)(aaaa)` ###color(white)(aaaaa)### `color(white)(aaaaaa)` `1` `color(white)(aaaa)` `3` `color(white)(aaaa)` `3` `color(white)(aaaa)` `3` `color(white)(aaaa)` `0` `color(white)(aaaaa)` `color(red)(3)`

The remainder is also `=3`

The quotient is `=x^4+3x^3+3x^2+3x`