How can I solve this by long division? Can the remainder have a variable?

Question

(3x3-2x2-8)/(x2+5)

Can the remainder be -15x-18?
Why or why not?

Answer 1

It's hard to show long division here, but I will answer the other question...

 

Yes, the remainder can have a variable in it.

 

The rule is that the remainder can be a polynomial but it has to be of degree less than the degree of the denominator.

 

So in this case the denominator is a 2nd degree polynomial.  So remainder has to be a first degree polynomial or a constant.

 

And I worked it out on paper, and I got the same remainder as you.  So you probably have the answer right!