Question

How do you use the remainder theorem and synthetic division to find the remainder when `2x^3-7x^2 div x-5`?

Answer 1

The remainder is `=75`

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)=2x^3-7x^2`

and `(x-5)`

`f(5)=2*125-175=250-175=75`

The remainder is `=75`

We now perform the synthetic division

`color(white)(aaaa)` `5` `color(white)(aaaa)` `|` `color(white)(aaaa)` `2` `color(white)(aaaa)` `-7` `color(white)(aaaa)` `0` `color(white)(aaaa)` `0`

`color(white)(aaaa)` `color(white)(aaaaa)` `|` `color(white)(aaaaa)` `color(white)(aaaa)` `10` `color(white)(aaaa)` `15` `color(white)(aaaa)` `75`

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

`color(white)(aaaa)` `color(white)(aaaaa)` `color(white)(aaaaaa)` `2` `color(white)(aaaa)` `3` `color(white)(aaaa)` `15` `color(white)(aaaa)` `color(red)(75)`

The remainder is also `=75`

The quotient is `=2x^2+3x+15`