(a)

Determine the zeros of the function and their multiplicity:

The polynomial function is .

Definition of real zeros:

If is a function and is a real number for which , then is called a real zero of .

From the definition of real zeros :.

and

and

and .

The real zeros of this polynomial function are and .

The definition of zeros of multiplicity :, the exponent of factor is .

is a zero of multiplicity because the exponent on the factor is .

is a zero of multiplicity because the exponent on the factor is .

(b)

Determine whether the graph crosses or touches the -axis at each -intercept.

is an -intercept of function .

Therefore, the -intercepts are and .

The zero is a zero of multiplicity , so the graph of touches the -axis at  .

The zero is a zero of multiplicity , so the graph of crosses the -axis at  .

(c)

Determine the behavior of the graph of near each -intercept:

The -intercepts are and .

Near :

.

Near :

.

(d)

Determine the maximum number of turning points on the graph of the function:

The polynomial function is .

Degree of the polynomial function is .

Maximum number of turning points is  .

.

At most turning points .

(e)

Determine the end behavior of the graph of the function:

The polynomial function is .

Expand the polynomial.

.

The polynomial function is of degree .

The function behaves like for large values of .

(a)

, multiplicity is .

, multiplicity is .

(b) The graph touches the -axis at and crosses at  .

(c)

Near : ,

Near : .

(d) The maximum number of turning points are .

(e)  The function behaves like for large values of .