Does a polynomial minus the tangent at a certain point have a double root? Given: $P(x)$ is a polynomial of at least the second degree and $L(x)$ is the tangent to $P(x)$ at $x=a$. 
Questions: Can one then say that $P(x)-L(x)$ has a double root at $x=a$? If so, why? If not, why not?

You can always rewrite your polynomial as $P(x) = P(a) + P'(a) (x-a) + \frac{P''(a)}{2}(x-a)^2 + \dots $. Using this form, you can write $L(x)$ as $L(x) = P(a) + P'(a)(x-a)$. Combining these two formulas we get that
$P(x) - L(x) = (x-a)^2 \cdot (\frac{P''(a)}{2} + \dots )$, which suggests that at point $a$ function $P(x)-L(x)$ has at least double root: it is double root when $P''(a) \neq 0$. In other cases the multiplicity of root depends on which derivative has first non-zero value.
