Exercises with Answers

Section 6.1 Polynomial Functions 559
6.1 Exercises
In Exercises 1-8, arrange each polynomial
in descending powers of x, state the
degree of the polynomial, identify the leading
term, then make a statement about
the coefficients of the given polynomial.
10.
y
5
1. p(x) = 3x − x 2 + 4 − x 3
2. p(x) = 4 + 3x 2 − 5x + x 3
x
5
3. p(x) = 3x 2 + x 4 − x − 4
4. p(x) = −3 + x 2 − x 3 + 5x 4
5. p(x) = 5x − 3 2 x3 + 4 − 2 3 x5
6. p(x) = − 3 2 x + 5 − 7 3 x5 + 4 3 x3
7. p(x) = −x + 2 3 x3 − √ 2x 2 + πx 6
11.
y
5
8. p(x) = 3+ √ 2x 4 + √ 3x−2x 2 + √ 5x 6
x
5
In Exercises 9-14, you are presented with
the graph of y = ax n . In each case, state
whether the degree is even or odd, then
state whether a is a positive or negative
number.
9.
y
5
12.
y
5
x
5
x
5
1
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Version: Fall 2007

560 Chapter 6 Polynomial Functions
13.
16.
y
5
y
x
5
x
14.
17.
y
5
y
x
5
x
In Exercises 15-20, you are presented
with the graph of the polynomial p(x) =
a n x n + · · · + a 1 x + a 0 . In each case, state
whether the degree of the polynomial is
even or odd, then state whether the leading
coefficient a n is positive or negative.
15.
18.
y
x
y
x
Version: Fall 2007

Section 6.1 Polynomial Functions 561
19.
y
21. p(x) = −3x 3 + 2x 2 + 8x − 4
22. p(x) = 2x 3 − 3x 2 + 4x − 8
23. p(x) = x 3 + x 2 − 17x + 15
x
24. p(x) = −x 4 + 2x 2 + 29x − 30
25. p(x) = x 4 − 3x 2 + 4
20.
y
26. p(x) = −x 4 + 8x 2 − 12
27. p(x) = −x 5 + 3x 4 − x 3 + 2x
28. p(x) = 2x 4 − 3x 3 + x − 10
29. p(x) = −x 6 − 4x 5 + 27x 4 + 78x 3 +
4x 2 + 376x − 480
x
30. p(x) = x 5 −27x 3 +30x 2 −124x+120
For each polynomial in Exercises 21-
30, perform each of the following tasks.
i. Predict the end-behavior of the polynomial
by drawing a very rough sketch
of the polynomial. Do this without
the assistance of a calculator. The
only concern here is that your graph
show the correct end-behavior.
ii. Draw the graph on your calculator,
adjust the viewing window so that
all “turning points” of the polynomial
are visible in the viewing window,
and copy the result onto your
homework paper. As usual, label and
scale each axis with xmin, xmax, ymin,
and ymax. Does the actual end-behavior
agree with your predicted end-behavior?
Version: Fall 2007

562 Chapter 6 Polynomial Functions
6.1 Answers
1. p(x) = −x 3 −x 2 +3x+4, degree = 3,
leading term = −x 3 , “p is a polynomial
with integer coefficients,” “p is a polynomial
with rational coefficients,” or “p is
a polynomial with real coefficients.”
21. Note that the leading term −3x 3
(dashed) has the same end-behavior as
the polynomial p.
y
10
3. p(x) = x 4 + 3x 2 − x − 4, degree = 4,
leading term = x 4 , “p is a polynomial
with integer coefficients,” “p is a polynomial
with rational coefficients,” or “p is
a polynomial with real coefficients.”
−10
x
10
5. p(x) = − 2 3 x5 − 3 2 x3 +5x+4, degree =
5, leading term = − 2 3 x5 , “p is a polynomial
with rational coefficients,” or “p is
a polynomial with real coefficients.”
−10
p(x)=−3x 3 +2x 2 +8x−4
7. p(x) = πx 6 + 2 3 x3 − √ 2x 2 −x, degree =
6, leading term = πx 6 , “p is a polynomial
with real coefficients.”
9. y = ax n , n odd, a < 0.
11. y = ax n , n even, a > 0.
23. Note that the leading term x 3 (dashed)
has the same end-behavior as the polynomial
p.
y
−70
p(x)=x 3 +x 2 −17x+15
13. y = ax n , n odd, a < 0.
15. odd, positive
17. even, negative
x
−10 10
19. odd, positive
−70
Version: Fall 2007

Section 6.1 Polynomial Functions 563
25. Note that the leading term x 4 (dashed)
has the same end-behavior as the polynomial
p.
29. Note that the leading term −x 6 (dashed)
has the same end-behavior as the polynomial
p.
y
10
p(x)=x 4 −3x 2 +4
y
5000
−10
x
10
x
−10 10
−10
−5000
p
27. Note that the leading term −x 5 (dashed)
has the same end-behavior as the polynomial
p.
y
15
x
−10 10
−10
p(x)=−x 5 +3x 4 −x 3 +2x
Version: Fall 2007