If a polynomial \(f(x)\) has real coefficients, and has a nonzero constant term, then the number of positive real roots is the number of sign changes of the coefficients or a multiple of two less than that.

Likewise, look at \(f(-x)\) for the negative real roots.

This can be useful in showing the existence of a real root (e.g. if the number of sign changes is odd).

One application of this rule is the ability to find lower/upper bounds on real roots. Let \(p(x)\) be a polynomial. Use synthetic division to divide by \(x-c\), where \(c>0\). If the entries in the final row of the division are all positive, then there cannot be a root greater than \(c\).

We can illustrate this with an example. Let \(p(x)=2x^{3}+5x^{2}-8x-7\). Then:

It’s straightforward to see that the RHS is positive for all \(x>2\).

Similarly, if \(c<0\) then we can put a lower bound on the real roots.