Let $p(x)$ be any polynomial with real coefficients, and $d(x)$ a monic quadratic polynomial. By the division algorithm we may write $$p(x) = d(x) q(x) + r(x)$$ where $\deg r(x) < 2$. Here, $q(x)$ is the quotient and $r(x)$, the remainder, is at most linear.
Now if the divisor has two distinct real zeros -- that is, if $d(x) = (x-a)(x-b)$ for some $a \ne b$ -- then the quotient and remainder have a natural geometric interpretation:
- $r(x)$ gives the line that passes through the points $(a,p(a))$ and $(b, p(b))$.
- $q(x)$ gives the leading coefficient of the unique parabola that passes through the three points $(a, p(a))$, $(b, p(b))$, and $(x, p(x))$.
If the divisor has only one zero with multiplicity 2 -- that is, if $d(x) = (x-a)^2$ for some $a$ -- then there is a slightly different geometric interpretation:
- $r(x)$ gives the line that passes through $(a, p(a))$ and is tangent to $p(x)$ at that point
- $q(x)$ gives the leading coefficient of the unique parabola that passes through $(a, p(a))$ and $(x, p(x))$ and is tangent to $p(x)$ at $a$.
My question: Is there a geometric interpretation of the quotient and remainder in the case where $d(x)$ has no real roots?
(And before anybody says it in an answer: I know that if you pass to complex numbers then the first interpretation above holds. I'm looking for a geometric interpretation of what's happening in $\mathbb{R}^2$. Such an interpretation -- if one exists at all -- would presumably not say anything about the roots of $d(x)$, but it might conceivably make reference to the real part of those roots, for example. )