Need two examples using this chart.

We will use Descarte's Rule to solve the following question:

It says that the number of sign changes in the function f(x) tells the maximum number of positive roots that could exist.

Similarly the number of sign changes in the function f(-x) tells the maximum  number of negative roots that could exist.

Now let us consider a example as:

We know that the total number of zeros of a polynomial function is always equal to the degree of the polynomial.

This is a polynomial function of degree 3 hence it has total 3 zeros.

Now f(x) has total 2 sign changes first from + to - and then from - to +.

Hence atmost 2 positive real zeros are possible.

Also f(-x)=-x^3-x^2+5

This function has only one sign change i.e. from - to +.

Hence atmost 1 negative real roots are possible.

Also on solving the cubic equation we got that we have one real zeros and 2 complex zeros.

In table we could write as:

total zeros:                 3    (-1.4334 ,  1.2167-1.4170 i ,  1.2167+1.4170 i)

No. of positive :          None

real zero

No. of negative :          1  (-1.4334)

real zero

Complex zero :              2  (1.2167-1.4170 i and  1.2167+1.4170 i)

This is a polynomial function of degree 4 hence it has total 4 zeros.

Now f(x) has total 1 sign changes first from + to - .

Hence atmost 1 positive real zeros are possible.

Also f(-x)= x^4-x^3-1

This function has only one sign change i.e. from + to -.

Hence atmost 1 negative real roots are possible.

Also on solving the cubic equation we got that we have two real zeros and 2 complex zeros.

In table we could write as:

total zeros:                  4    (-1.3803, 0.81917 , -0.21945-0.91447 i ,    

                                                 ,   -0.21945+0.91447 i)    

No. of positive :          1 (0.81917)

real zero

No. of negative :          1  (-1.3803)

real zero

Complex zero :              2 (-0.21945-0.91447 i , -0.21945+0.91447 i)