Fundamental Theorem of Algebra : A polynomial of degree n can have at most n distinct real roots.

Descartes' Rule of Signs : It tells us that the number of positive real roots in a polynomial function f(x) is the same or less than by an even numbers as the number of changes in the sign of the coefficients. The number of negative real roots of the f(x) is the same as the number of changes in sign of the coefficients of the terms of f(-x) or less than this by an even number.

Example : Determine the number of positive and negative real roots for the given function f(x) = x^5 + 4x^4 − 3x^2 + x − 6

Our function is arranged in descending powers of the variable, if it were not we would have to do that as a first step. Second we count the number of changes in sign for the coefficients of f(x).

Here are the coefficients of our variable in f(x): 1 + 4 − 3 + 1 − 6

Our variables goes from positive(1) to positive(4) to negative(-3) to positive(1) to negative(-6).

Between the first two coefficients there are no change in signs but between our second and third we have our first change, then between our third and fourth we have our second change and between our 4th and 5th coefficients we have a third change of coefficients.

Descartes´ rule of signs tells us that the we then have exactly 3 real positive roots or less than by it an even number. Hence our number of positive roots must then be either 3 or 1 ( = 3 - 2).

In order to find the number of negative zeros we find f(-x) and count the number of changes in sign for the coefficients:

googletag.cmd.push(function() { googletag.display('div-gpt-ad-1492166336268-0'); });f(−x) = (−x)^5 + 4(−x)^4−3(−x)^2 + (−x) − 6 = −x^5 + 4x^4−3x^2 − x − 6

Here we can see that we have two changes of signs, hence we have two negative roots or less than it by an even number.

Totally we have 3 or 1 positive roots or 2 or 0 negative roots.

Descartes' Rule of Signs does not provide the exact number of positive and negative real roots.

Refer : https://www.mathplanet.com/education/algebra-2/polynomial-functions/descartes-rule-of-sign