Descartes's rule of signs

The Descartes's rule of signs enables us to determine the number of positive and negative roots of a polynomial function, however, it is not able to give the exact number of real and complex roots a polynomial function may have.

The Descartes's rule of signs says that the number of times that the sign on consecutive terms in a polynomial changes from + to - or from - to +, is the number of positive roots of the polynomial function. On the other hand, if the sign on the variable in a polynomial function is changed, then number of times sign changes occur throughout the polynomial is the number of negative roots that the polynomial possesses.


Positive and negative roots

For example, for the polynomial x^4 + 2x^2 - x + 1, there are two sign changes, one from positive 2x^2 to negative -x, and the other from negative -x to positive 1. Thus, the polynomial  x^4 + 2x^2 - x + 1 has two positive roots.

On changing the sign of the variable in the polynomial, you get the polynomial (-x)^4 + 2(-x)^2 - -x + 1, which simplifies to x^4 + 2x^2 + x + 1. There are no sign changes in this polynomial, since all terms are positive. Thus, the polynomial   x^4 + 2x^2 + x + 1 does not have any negative roots.

Similarly, the polynomial 3x^6 - x^5 -  2x^4 + x^3 - x^2 + x - 4 has five sign changes:

1. From positive 3x^6 to negative -x^5
2. From negative 2x^4 to positive x^3
3. From positive x^3 to negative x^2
4. From negative x^2 o positive x
5. From positive x to negative 4

Thus, the polynomial 3x^6 - x^5 -  2x^4 + x^3 - x^2 + x - 4 has five positive roots. On changing the sign of the variable,

3(-x)^6 - (-x)^5 -  2(-x)^4 + (-x)^3 - (-x)^2 + (-)x - 4

which simplifies to

3x^6 + x^5 -  2x^4 - x^3 -x^2 -x - 4 

Now in this polynomial, only one sign change is occurring, that from positive x^5 to negative -2x^4. So, there is only one negative root of the polynomial 3x^6 - x^5 -  2x^4 + x^3 - x^2 + x - 4.

Complex roots

Complex roots are those roots that include imaginary numbers in them, for example, 3 + i and 1 - 2i. A polynomial can have complex roots as well as positive roots. If a polynomial has complex roots, they occur in conjugate pairs, that is, a polynomial can have 0, 2, 4 or any even number of complex roots. A polynomial can not have 1, 3, 5, .. or an odd number of complex roots.

By using the Descartes's rule of signs, you can only get an estimate of how many complex, positive, or negative roots a polynomial may have.

The number of positive roots of a polynomial include the complex roots. So, if a polynomial has 5 positive roots, then out of those 5 positive roots, either 0, 2 or 4 may be complex and the others are real positive roots.