**Introduction** Suppose that *y* = *f*(*x*) is a polynomial function with real coefficients and is arranged in the usual manner of descending powers of *x*. From this form it is possible to determine the maximum number of positive zeros and the maximum number of negative zeros by examining the variations of sign in *f*(*x*). We say that a **variation of sign** occurs when two consecutive terms have opposite signs. For example, in the polynomial

there are three variations of sign: between the first and second terms, between the third and fourth terms, and between the fourth and fifth terms.

We state the following rule without proof. ...

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