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?

Let $$g(x) = P(x) - L(x)$$
Since $P(a) = L(a)$ (because tangent touches curve at $x = a$), we have 
$$g(a) = P(a) - L(a) = 0$$
Now, differentiate $g(x)$ to get:
$$g'(x) = P'(x) - L'(x)$$
At $x = a$, $L'(a) = P'(a)$ (because tangent) so
$$g'(a) = P'(a) - L'(a) = 0$$
Since $a$ is (at least) a double root of $g(a)$ if and only if $g(a) = 0, g'(a) = 0$, we deduce that $a$ is indeed (at least) a double root.
