College Algebra 9th txtbk

262 CHAPTER 4 POLYNOMIAL AND RATIONAL FUNCTIONS
SOLUTIONS (A) The real zeros are the x intercepts: 4, 2, 0, and 3.
(B) Note first that P(x) is a polynomial because it can be written in the form of Definition 1
(it is not necessary to actually multiply out P(x) to find that form). The zeros of P(x) are
the solutions to the equation P(x) 0. Because a product equals 0 if and only if one
of the factors equals 0, we can find the zeros by solving each of the following equations
(the last was solved using the quadratic formula):
x 4 0 (x 7) 3 0 x 2 9 0 x 2 2x 2 0
x 4 x 7 x 3i x 1 i
Therefore, the zeros of P(x), are 4, 7, 3i, 3i, 1 i, and 1 i. Only two of the
six zeros are real numbers and therefore x intercepts: 4 and 7.
MATCHED PROBLEM 1
(A) Figure 4 shows the graph of a polynomial function of degree 4. List its real zeros.
5
5
5
5
Z Figure 4
(B) List all zeros of the polynomial function
P(x) (x 5)(x 2 4)(x 2 4)(x 2 2x 5)
Which zeros of P(x) are x intercepts?
A point on a continuous graph that separates an increasing portion from a decreasing
portion, or vice versa, is called a turning point. The vertex of a parabola, for example, is
a turning point. Linear functions with real coefficients have exactly one real zero and no
turning points; quadratic functions with real coefficients have at most two real zeros and
exactly one turning point.
ZZZ EXPLORE-DISCUSS 1
Examine Figures 2(a), 2(b), 3, and 4, which show the graphs of polynomial functions
of degree 2, 2, 5, and 4, respectively. In each figure, all real zeros and all
turning points of the function appear in the given viewing window.
(A) Is the number of real zeros ever less than the degree? Equal to the degree?
Greater than the degree? How is the number of real zeros of a polynomial related
to its degree?
(B) Is the number of turning points ever less than the degree? Equal to the degree?
Greater than the degree? How is the number of turning points of a polynomial
related to its degree?