College Algebra 9th txtbk

SECTION 4–2 Real Zeros and Polynomial Inequalities 283
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(a) (b) (c)
Z Figure 6 Zeros of P(x) x 5 6x 4 4x 3 24x 2 16x 32.
MATCHED PROBLEM 4
Let P(x) x 5 6x 4 40x 2 12x 72. Find the smallest positive integer and the largest
negative integer that, by Theorem 1, are upper and lower bounds, respectively, for the real
zeros of P(x). Approximate the zeros to two decimal places, using maximum or minimum
commands to approximate any zeros at turning points.
Z Polynomial Inequalities
We can apply the techniques we have introduced for finding real zeros to solve polynomial
inequalities. Consider, for example, the inequality
x 3 2x 2 5x 6 7 0
The real zeros of P(x) x 3 2x 2 5x 6 are easily found to be 2, 1, and 3. They
partition the x axis into four intervals
(, 2), (2, 1), (1, 3),
and
(3, )
On any one of these intervals, the graph of P is either above the x axis or below the x axis,
because, by the location theorem, a continuous function can change sign only at a zero.
One way to decide whether the graph of P is above or below the x axis on a given
interval, say (2, 1), is to choose a “test number” that belongs to the interval, 0, for example,
and evaluate P at the test number. Because P(0) 6 0, the graph of P is above the
x axis throughout the interval (2, 1). A second way to decide whether the graph of P is
above or below the x axis on (2, 1) is to simply inspect the graph of P. Each technique
has its advantages, and both are illustrated in the solutions to Examples 5 and 6.
EXAMPLE 5 Solving Polynomial Inequalities
Solve the inequality x 3 2x 2 5x 6 0.
SOLUTION
Let P(x) x 3 2x 2 5x 6. Then
P(1) 1 3 2(1 2 ) 5 6 0
so 1 is a zero of P and x 1 is a factor. Dividing P(x) by x 1 (details omitted) gives
the quotient x 2 – x 6. Therefore,
P(x) (x 1)(x 2 x 6) (x 1)(x 2)(x 3)
The zeros of P are 2, 1, and 3. They partition the x axis into the four intervals shown in
the table on page 284. A test number is chosen from each interval as indicated to determine
whether P(x) is positive (above the x axis) or negative (below the x axis) on that interval.