P(x)/q(x)= c(x)+r(x)/q(x). what if r(x)=0?

"You can write a rational function after a long division by the product of the two polynomials plus the remainder divided by the denominator." - Well, not quite. It's the quotient of the two polynomials plus the remainder divided by the denominator.

"Can you show me why this is true?"

In primary school you have learned how to divide an integer by another integer. Lets say,

162 DIV 12 = 13 , remainder 6 , where DIV is the integer division operator. You could also write

162 / 12 = 13 + 6/12. This looks already somewhat similar to

P(x) / Q(x) = C(x) + R(x) / Q(x).

Now, remind yourself of what the representation of a number in our positional notation system means, if we write it down in its structure
162 / 12 would look like this:

(x^2 + 6x + 2) : (x + 2), with the base 10 as x:

(1 * 10^2 + 6 * 10^1 + 2 * 10^0) : (1 * 10^1 + 2 * 10^0).

"What if R(x)=0?"

Then

P(x) / Q(x) = C(x) + R(x) / Q(x) reduces to

P(x) / Q(x) = C(x) + 0, just as in the division of numbers, e.g. 156 / 12 = 13.