Is this correct?

Question

In synthetic division and long division is it possible for there to be a zero remainder or no remander at all so
my answer w/ out a remainder would look like -1x+10x-25
and with a remainder it would look like 3x+2x+5-23/x+5

I really am not sure my question is really can a problems answer have no remainder at all?

Answer 1

Sure. For example, if you divide (x² - 3x + 2) by (x -1) you will get (x - 2) with no remainder. This is an indication that (x - 1) is a factor of (x² - 3x + 2).

On the other hand, if you divide (x² - 3x + 2) by (x + 1), you will get (x - 4) + 6/(x+1)

Answer 2

In synthetic division ( you are dividing by (x-a)) the remainder is always a number.If zero" a "is a root of the polynomial
In the division of two polynomials with degree m and n with
m>n the remainder is a polynomial with degree h It can be a number including zero(in this case the polynomial is of degree zero)
In any case the remainder always exists