WOLFRAM|DEMONSTRATIONS PROJECT

The Number of Distinct Real Roots of a Real Polynomial

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coefficients
6
x
1
5
x
0
4
x
-6
3
x
0
2
x
9
x
0
free
-4
polynomial : -4+9
2
x
-6
4
x
+
6
x
interval : {-3,3}
total number of distinct real roots : 4
number of real roots inside the interval : 4
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The problem of deciding if a real polynomial has a real root and if so, how many, has a long history going back to Descartes's Law of Signs. In more recent times substantial progress has been made in this area by several Chinese mathematicians. Here, some of their work is demonstrated: the computation of the number of distinct roots of a polynomial as well as the number of distinct roots lying in a given interval. The method is in the spirit of Descartes's Law of Signs and does not involve any root finding or root isolation.
In this Demonstration you can vary the coefficients of a polynomial of degree 6 and the interval in which the roots are sought. The interval is varied by moving its end points with the mouse.