If $P(x)$ is a quartic polynomial and if $l$ is the line that is tangent to the graph $P(x)$ at two places and $l(x)$ is the function defined by line $l$, how do I know for sure that $P(x)-l(x)$ is then tangent to the x-axis at two points?
The original problem is as follows - if it helps you understand my question better: Show that if $P(x)$ is a quartic polynomial then there exists at most one line $\ell$ that is tangent to the graph of $P(x)$ at two places.