PRACTICE
1. Which of the following statements best
describes the zeros of the quartic polynomial
function graphed?
y
5. The graph of a polynomial function passes
through the x-axis three times and the y-axis
once. How many real zeros does it have?
10
5
–5
–4
–3
–2
–1
1
–5
2
3
4
5
x
Exercises 6–9: Use each graph to determine the
number of real zeros of the function.
–10
A. There are 4 real zeros.
B. There are 4 complex zeros.
C. There are 3 real zeros and
1 complex zero.
D. There are 2 real zeros and
2 complex zeros.
–3
–1
3
x
–3
7.
y
10
5
1 2 3 4 5 6 7 8 9 10
–5
x
–4
–3
–2
–1
–5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
x
–10
–15
8.
y
2
150
100
50
–5
–4
–3
–2
3
0.12
4
9.
y
60
40
Which of his statements is true about P(x)?
32 Chapter 3: Polynomials
x
–150
21.6
A. The polynomial has a zero at x 5 3 and
x 5 4.
B. The polynomial has a zero between
x 5 3 and x 5 4.
C. The polynomial has factors x 2 0.12 and
x 1 1.6.
D. The polynomial has at least 2
x-intercepts.
–1
–50
–100
3. MP 2 Mark analyzes a polynomial P(x) and
makes the following table of values.
P(x)
2
–2
f(x) 5 2x 1 2x 24x 1 2
f(x) 5 x3 1 3x2 22x 1 1
f(x) 5 2x3 1 5x2 22x
f(x) 5 2x3 2 x2 1 3x 2 4
x
1
15
7
6
5
4
3
2
1
A.
B.
C.
D.
–2
–1
y
3
y
6.
2. Which of the following polynomial
functions does this graph best represent?
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
4. The graph of a polynomial function never
passes through the x-axis but passes through
the y-axis once. How many real zeros does
it have?
20
–5
–4
–3
–2
–1
–20
x
–40
–60
Practice Problems continue . . .
Practice Problems continued . . .
10. Use the graph of the polynomial function
f(x) 5 x3 1 x2 2 3x 2 3 to approximate its
real zeros, to the nearest tenth.
14. Find the cubic polynomial with leading
coefficient 1 that has the zeros shown in
the graph.
y
y
6
250
200
4
150
100
2
50
–3
–2
–1
1
2
3
x
–10 –8
–6
–4
–2
–2
–50
2
4
6
8
10
x
–100
–150
–4
–200
–250
–6
Exercises 11–13: Use iteration to determine the
real zeros, to the nearest tenth.
11. P(x) 5 x4 2 6x2 2 7
15. Find the quartic polynomial with leading
coefficient 1 that has the zeros shown in the
graph.
y
y
25
2500
20
2000
15
1500
10
1000
500
5
–5
–4
–3
–2
–1
1
–5
2
3
4
x
5
–10 –8
–1000
–1500
150
100
50
–3
–2
–1
–50
1
2
3
4
x
5
–100
–150
y
60
40
20
–4
–3
–2
–1
–20
–40
–60
1
2
2
4
6
8
10
x
16. A student sees a graph of a cubic polynomial
that passes through the x-axis three times.
She then factors the polynomial and
concludes it has the factors x2 1 x 1 1 and
x 2 1. Can she be right? Why?
17. A polynomial whose graph is a straight line
can have how many real zeros? Give an
example for each answer you state.
18. A local maximum is the point where the
graph of a polynomial function reaches its
highest value within a restricted area
or domain. For the polynomial
P(x) = x3 1 8x2 1 12x 2 6, a local maximum
occurs between x 5 25 and x 5 24.
13. P(x) 5 x4 2 9x2 2 52
–5
–2
–500
–15
y
–4
–4
–10
12. P(x) 5 x5 1 x4 2 11x3 2 11x2 2 12x 2 12
–5
–6
3
4
5
x
aUse iteration to find the coordinates of
the local maximum, to the nearest tenth.
bGraph the function for the domain
26 # x # 22, and label the local
maximum.
Practice Problems continue . . .
3.5 Finding Zeros of Polynomial Functions 33
Practice Problems continued . . .
19.aGiven the function y 5 x3 1 6x2 2 5x 2 4,
make a table of values of y for integer
values of x in the domain 23 # x # 3.
bBased on the values in your table,
how many times will the graph of the
­function cross the x-axis in the given
domain?
cGraph the function for the stated domain
and use it to check your answer to
part b.
20. MP 3 Explain how to find a zero of the
polynomial function P(x) if you know
P(1) 5 22 and P(2) 5 5.
21. Write a polynomial that has no real zeros.
22. Eric says because the graph of a polynomial
function crosses the x-axis three times, it has
three real zeros and must be a third-degree
polynomial. What mistake did Eric make in
his reasoning?
Exercises 23–24: Find the real zeros of the
polynomial, to the nearest tenth over the interval
given, using a calculator.
23.f(x) 5 x4 2 2x3 2 x2 2 2x 1 1 from x 5 23
to x 5 3
24.f(x) 5 x4 1 x3 2 7x2 2 8x 2 2 from x 5 23
to x 5 3
Exercises 25–32: Use a calculator to graph the
function and state its real zeros.
25.f(x) 5 2x3 1 x2 1 x 2 1 from x 5 22 to x 5 2
26.f(x) 5 x4 2 x3 2 x 1 1 from x 5 22 to x 5 2
27.f(x) 5 x3 1 3x2 2 x 2 3 from x 5 24 to x 5 4
28. f(x) 5 x3 2 x2 2 4x 1 4 from x 5 23 to x 5 3
34 Chapter 3: Polynomials
29. f(x) 5 2x3 1 3x2 2 2x 2 3 from x 5 22 to
x52
30. f(x) 5 x4 1 x3 2 6x2 2 4x 1 8 from x 5 23 to
x53
31. f(x) 5 2x4 2 11x3 1 16x2 2 x 2 6 from
x 5 24 to x 5 4
32. f(x) 5 4x3 1 4x2 2 x 2 1 from x 5 22 to
x 5 2.
33. What are the zeros of P(x) 5 x3 1 x2 2 4x 2 4?
34. What are the zeros of P(x) 5 2x3 2 5x2 2 2x 1 5?
Round to the nearest tenth.
35. What are the zeros of P(x) 5 2x3 1 3x2 2 11x 2 6?
Round to the nearest tenth.
36. Investigate the properties of the graphs of
fifth-, sixth-, seventh-, and eighth-degree
polynomials. How are these graphs
related to those of third- and fourth-degree
polynomials?
37. A quartic polynomial can have zero real
roots, and its graph will never cross
the x-axis. Is this also true for a cubic
polynomial? Explain your reasoning.
38. MP 2, 7 A third-degree polynomial has
three real roots. The roots are consecutive
odd integers, and their sum is three.
aFind the roots, and write an equation for
the polynomial function with these roots
and a leading coefficient of 1.
bGraph the polynomial function you wrote
in part a for the domain 22 # x # 4.