Polynomial long division

Question

How do we solve this:

(y^2 - x y + x^2)/(-y^3 + x^2 y^2 + y^2 x - x^3 y - y x^2 + x^4)

Answer 1

switch the dividend and divisor, (switch numerator and denominator) then do long division to get -y+ x^2

now find the inverse: 1/(-y+x^2) = the quotient of the original problem. that's the solution.

try a simpler problem to see more clearly what's going on

find a/ab, a divided by ab

a/ab = 1/b

switch the dividend and divisor (switch numerator and denominator) ab/a = b, now find the inverse of b to get : 1/b that's the solution to a/ab

do the same with the quadratic divisor and the cubic dividend. switch them, then it divides evenly getting a quotient of -y+x^2. Now flip it to get the inverse: 1/(-y+x^2) that's the answer

in the given complicated problem a= the quadratic divisor y^2 - xy + x^2 and ab= the six term cubic polynomial and b=-y+x^2, then 1/b = 1/(-y+x^2) = the solution