Question

How do you use polynomial division to determine whether x=3 is a root of `x^3+x^2+15-x`?

Answer 1

Divide `(x^3 + x^2 -x + 15) -: (x-r)` with long polynomial division.

If you can divide without a remainder, `x = r` is a root.
If you have a remainder, it's not a root.

Explanation:

Well, just to check if `x = 3` is a root of your term, it would be easier to plug `x = 3` and see if the result is `0`. :-)

But of course, it is also possible to determine the outcome with polynomial division.

To do so, you must divide `(x^3 + x^2 -x + 15)` by `(x-3)`.

I know that in some countries, a different notation for long division is being used but I will use the one that I'm familiar with and I hope that it will be no problem for you to rewrite it in your notation if necessary.

`color(white)(xx)(x^3 + x^2 color(white)(xx) - x color(white)(xx) + 15) : (x-3) = x^2 + 4x + 11`
`color(white)(i)-(x^3 - 3x^2)`
`color(white)(xxx)color(white)(xxxxxx)/color(white)(x)`
`color(white)(xxxxxxx) 4 x^2 color(white)(x)- x`
`color(white)(xxxxx) -(4 x^2 - 12x)`
`color(white)(xxxxxxx)color(white)(xxxxxxxx)/color(white)(x)`
`color(white)(xxxxxxxxxxxx) 11x color(white)(x) + 15`
`color(white)(xxxxxxxxxx) -(11x color(white)(x) - 33)`
`color(white)(xxxxxxxxxxxx)color(white)(xxxxxxxx)/color(white)(x)`
`color(white)(xxxxxxxxxxxxxxxxxx) 48`

At the end of the computation, there is the remainder `48`.

As you have a remainder, it means that `x = 3` is not a root of `x^3 + x^2 -x + 15`.