# Roots and Tangents

The Random Walk blog by Mr Chase brought to my attention the following fact:

The line tangent to the graph of a 3rd degree polynomial at the point midway between two roots passes through the third root.

The applet below serves a dynamic illustration

This is an exciting property of third degree polynomials which does not appear to generalize to polynomials of a higher degree. However, there is a marvelous generalization due to Qiaochu Yuan, a moderator at the stackexchange.com.

Let $f(x)=(x-a)g(x)$ where $g(x)$ is any differentiable function, and let $r$ be such that $g'(r)=0$. Then the tangent line to $y=f(x)$ at $(r,f(r))$ intersects the $x$-axis at $(a,0)$.

### Proof

First, $f'(x)=(x-a)g'(x) + g(x)$ so that $f'(r)=g(r).$ Form this we obtain an equation for the tangent at $x=r$:

$t(x) = (x - r)g(r) + f(r)$

which, for $x=a$, gives

$t(a) = (a - r)g(r) + f(r) = -(r - a)g(r) + f(r) = 0$

because $f(r)=(r-a)g(r)$.

When $g$ is a second degree polynomial, say $g(x)=(x-b)(x-c)$, then $g'(x)=2x-(b+c)$ so that $\displaystyle g'(\frac{a+b}{2})=0.$

Copyright © 1996-2018 Alexander Bogomolny

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