I was reading about polynomials and long division at wikipedia, and came over this part.
Polynomial long division can be used to find the equation of the line that is tangent to the graph of the function defined by the polynomial $P(x)$ at a particular point $x = r$.
If $P(x)$ divided by $(x - r)^2$ leaves a remainder of $R(x)$ then the equation of the tangent line to $P(x)$ at $x = r$ is $y = R(x)$ (regardless of whether or not $r$ is a root of the polynomial).
Can someone give me an intuitive explenation of why this is true, or a proof? Googling gave me nothing, and a few drawings of using $(x-a)(x-\Delta x)$ seemed to have something to do with the line from $a$ to $b$. Now letting $\delta x \to a$ would give the tangent. Is this correct?
I do not quite see why dividing $P(x):(x-a)(x-\Delta x)$ would give me a line through $a$ and $\Delta x$ though..
Any help, suggestions or tips is very welcome =)