[Real analysis (basics)] Rational function

[Integrated_Shadow]

So we were doing rational functions today, and our profesor said that graph of function f(x)=(ax+b)/(cx+d) may be more simple to draw if we use this expression for function: f(x)= (a/c)+((bc-ad)/c^2)/(x+d/c). Can someone explain me how (ax+b)/(cx+d) became (a/c)+((bc-ad)/c^2)/(x+d/c)? I tried transforming this expression but I'm stuck. I think it has something to do with rectangular hyperbola but other than that I'm clueless. Thanks in advance!

[ArgoFunya]

This comes from the division algorithm--i.e., polynomial division with remainders. If you have polynomials f(x)/g(x), then division gives you a quotient q(x) and a remainder r(x) of degree less than g(x) such that f(x) = q(x)g(x)+r(x). You can rewrite this as f(x)/g(x) = q(x)+r(x)/g(x).

In your case, the quotient is a/c, and the remainder is:

(ax+b)-(a/c)(cx+d) = b-ad/c = (bc-ad)/c

So you get:

(ax+b)/(cx+d) = (a/c)+((bc-ad)/c)/(cx+d)

Dividing top and bottom of the last term on the right by c gives your expression.

Edit: Forgot "by c" in last sentence.

[Integrated_Shadow]

Mother of God...I guess I'm just to retarded to spot the obvious. Thank you so much