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\disleft 38pt::
{\bf Exercise in recursion.}

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Let $p(x)$ be a polynomial, $p'(x)$ its derivative. Let $r↓1,r↓2,\ldots,r↓k$
be the real roots of~$p'(x)$ in increasing order, and define $r↓0=-∞$,
$r↓{k+1}=+∞$. If $p(r↓1)$ and~$p(r↓{i+1})$ have the same sign, $p(x)$~has no
real roots in the interval $(r↓i,r↓{i+1})$; if they have opposite signs,
$p(x)$~has exactly one root in $(r↓i,r↓{i+1})$, and the root is readily
found by the bisection algorithm. Design a recursive procedure to find
all the  real roots of a polynomial based on the above.

\disleft 38pt::
Simplify by getting the real roots in a finite interval.



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\copyright 1984 Robert W. Floyd

First draft (not published) December 4, l980.

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