Is the function below a rational function?

Question

Is the function below a rational function? If it is, write it in reduced form as a ratio of polynomials p(x)/q(x). If it is not, enter NONE in both blanks. Do not leave any blanks empty.

x^2/x-1 - 6/x-2

Answer 1

x^2/x-1 - 6/x-2 is the sum of two rational functions, so it is also a rational function.

One computes the minimum common denominator to bring that into a p(x)/q(x) form:

x^2/x-1 - 6/x-2 = (x^3 -2x^2 -6x +6)/(x-1)(x-2)

Answer 2

x^2/x-1 - 6/x-2 = (x-2)×x²/(x-1)/(x-2) -6(x-1)/(x-2)/(x-1)

= [(x³ -2x²) -6x+6)]/[(x-1)(x-2)]

= ( x³ -2x² -x + 6)/(x² -3x +2)

which has the form p(x)/q(x), so it is a rational function.