The polynomial function is .

Step 1:

Find the total number of zeros.

Since the polynomial function has degree , the function has zeros.

Step 2:

Find the type of zeros.

Examine the number of sign changes for and .

The possible number of positive zeros of polynomial function is the number of sign changes of the coefficients of or that number minus even number.

.

Since there are sign changes in the the possible number of positive zeros of polynomial function is .

.

Since there are sign change in the the possible number of negative zeros of polynomial function is .

Hence, by Descartessign rule, the maximum number of zeros is .

Step 3:

Find the real zeros list some possible values and then use synthetic substitution to evaluate for real values of .

The depressed polynomial is .

The depressed polynomial is .

The quadratic function is .

                 (Quadratic formula)

        (Substitute and )

             (Simplify)

                             (Subtract and simplify)

and        (Separate two roots)

and             (Simplify)

The function has zeros at and .

Check solution for -values.

Graph:

Graph the equation .

Observe the graph:

The graph crosses the - axis at and .

Therefore, the value of is .

The function has real zeroes and imaginary zeroes.

Zeroes of the function are  and .