Leading term test for polynomial end behavior :

The end behavior of any non-constant polynomial function can be described in one of the following four ways, as determined by the degree of the polynomial and its leading coefficient .

For odd and positive :

and .

For odd and negative :

and .

For even and positive :

and .

For even and negative :

and .

A polynomial function of degree has at most distinct real zeros and at most turning points.

Multiplicity :

How many times a particular number is a zero for a given polynomial is known as multiplicity.

(a)

The polynomial function is .

The leading term is .

The degree is and the leading coefficient is .

Since the degree is (even) and the leading coefficient is (negative), the end behavior of the graph of is and .

(b)

The function is .

The degree of the function is .

Since the degree of the function is (), the function has at most () distinct real zeros.

Find the real zeros by equating to zero.

Solve the equation .

Apply zero product property.

Thus, the function has two distinct real zeros at and .

And the zeros at and has multiplicity .

(c)

The function is .

Make the table of values to find ordered pairs that satisfy the function.

Choose values for that fall in the intervals determined by the zeros of the function.

Interval	value in the interval

(d)

Graph :

(1) Draw a coordinate plane.

(2) Plot the points obtained by above table.

(3) The end behavior of the function tells that the graph of the function is eventually falls to the left and to the right.

(4) The graph does not crosses at repeated zeros, i.e, and , because its multiplicity is even.

(5) Draw a continues curve through the points.

.

(a) The end behavior of the graph of is and .

(b)

The function has two distinct real zeros at and and the multiplicity is .

(c)

The additional points are , and .

(d)

Graph of is :

.