Check out this dialogue between two calculus students (based on a true story):

Devyn
Riley, I’m wondering about polynomials.
Riley
What about them?
Devyn
If we have and , I know how to add them. I know how to subtract them. I know how to multiply them. I even know how to factor each of them.
Riley
Sure. To add or subtract, we combine like terms. To multiply, we use distribution or FOIL.
Devyn
But we can’t just divide by and arrive at a polynomial! If we divide polynomials, we don’t necessarily end up with another polynomial.
Riley
Sometimes we can, though. divided by is basically just the polynomial .
Devyn
But if we don’t get a polynomial back, what kind of function DO we get?
If we divide by , do we end up with a polynomial?
yes no
If we divide by , do we end up with a polynomial?
yes no