Will it divide?

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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 $x^2+x$ and $x^2-4$ , 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 $x^2+x$ by $x^2-4$ and arrive at a polynomial! If we divide polynomials, we don’t necessarily end up with another polynomial.
Riley: Sometimes we can, though. $x^2 + x$ divided by $x$ is basically just the polynomial $x+1$ .
Devyn: But if we don’t get a polynomial back, what kind of function DO we get?

If we divide $x^3 + 3x^2 + 3x + 1$ by $x+1$ , do we end up with a polynomial?

yes no

Remember our multiplication formula $(a+b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3$ . If we set $a = x$ and $b = 1$ , this tells us that $x^3 + 3x^2 + 3x + 1$ can be rewritten as $(x + 1)^3$ .

If we divide $x^3 + 3x^2 + 3x + 4$ by $x+1$ , do we end up with a polynomial?

yes no

We saw above that $x^3 + 3x^2 + 3x + 1$ can be rewritten as $(x + 1)^3$ . That means $x^3 + 3x^2 + 3x + 4 = (x+1)^3 + 3$ , so $\displaystyle \dfrac {x^3 + 3x^2 + 3x + 4}{x+1} = x^2 + 2x + 1 + \dfrac {3}{x+1}$ , which isn’t a polynomial.