Q:  Use polynomial long division to divide:  x²/(1-3x)

A:  OK… Trying to format this to look like a long division problem:

-3x + 1 | x²

(1) Ask yourself: how many times does -3x go into x²?  The answer is -(1/3)x, so we put that above:

………. -(1/3)x

-3x + 1 | x²

Now, we multiply -(1/3)x by -3x + 1 and bring it down like so:

………. -(1/3)x

-3x + 1 | x²

………….x² – (1/3)x

OK, now we subtract red from red to get blue:

………. -(1/3)x

-3x + 1 | x²

…………1(x² – (1/3)x)

………………….. (1/3)x

Now we ask ourselves:  How many times does -3x go into (1/3)x… The answer to that is -(1/9) times:

………. -(1/3)x – (1/9)

-3x + 1 | x²

…………1(x² – (1/3)x)

………………….. (1/3)x

Now, multiply -1/9 times -3x + 1 and bring down like so to get blue:

………. -(1/3)x – (1/9)

-3x + 1 | x²

…………1(x² – (1/3)x)

…………………….. (1/3)x

……………………..1/3x – 1/9

Subtract blue from blue to get purple:

………. -(1/3)x – (1/9)

-3x + 1 | x²

…………1(x² – (1/3)x)

…………………….. (1/3)x

……………………..-(1/3x – 1/9)

………………………………….1/9

And, we stop because -3x cannot go into 1/9…

So,

x²/(1-3x) = -(1/3)x – (1/9) + (1/9)/(-3x + 1) = -1/3x – 1/9 + 1/(-27x + 9).