Question

Need two examples using this chart. PLEASE!

Answer 1

Answer:

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

• Work area to determine possible positive real roots:

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

• Work area to determine possible negative real roots:

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:

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

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)

2) f(x)=x^4+x^3-1

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)