Question

In (8b+2b^3)/(b-1), Do you have to put a zero coefficient in it?

Answer 1

yes, if you want to perform the algebraic long division, or synthetic division.

arrange it like
2b^3 + 0x^2 + 8b ÷ (b-1)

with either method you should end up with

2b^2 + 2b + 10 with a remainder of 10

Answer 2

Are you sure it isn't 10b?

Answer 3

I am sure:

2b^3 + 0x^2 + 8b ÷ (b-1)
= (2b^2 + 2b + 10) + 10/(b-1)


check :
add the terms in my answer,

(2b^2 + 2b + 10) + 10/(b-1)
= [(2b^2 + 2b + 10)(b-1) + 10]/(b-1)
= (2b^3 + 8b) ÷ (b-1) or the original question.

another way to check answers like this is to take any value of b, say b=2
and sub it in the original and in the answer.

If you get the same result, there is a very good probablility that the question was done correctly.

This does not prove that it is right, but if you don't get the same result, then you know for sure that your answer is wrong.