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 positive odd number.

.

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

Since there are sign changes in the the impossible number of negative zeros of polynomial function is .

Hence, by Descartes sign 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 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 cross the - axis at ,  and

Therefore, the value of is and .

The function has number of real zeros.

Zeroes of the function are ,  and .