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 \(\displaystyle g'(\frac{a+b}{2})=0.\)

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