3.4 The Fundamental Theorem of Algebra
What you should learn
䊏 Use the Fundamental Theorem of Algebra
to determine the number of zeros of a
polynomial function.
䊏 Find all zeros of polynomial functions,
including complex zeros.
䊏 Find conjugate pairs of complex zeros.
䊏 Find zeros of polynomials by factoring.
Why you should learn it
Being able to find zeros of polynomial
functions is an important part of modeling
real-life problems.For instance,Exercise 63 on
page 297 shows how to determine whether a
ball thrown with a given velocity can reach a
certain height.
Jed Jacobsohn/Getty Images
Section 3.4 The Fundamental Theorem of Algebra 291
The Fundamental Theorem of Algebra
You know that an nth-degree polynomial can have at most n real zeros. In the
complex number system, this statement can be improved. That is, in the complex
number system, every nth-degree polynomial function has precisely n zeros. This
important result is derived from the Fundamental Theorem of Algebra, first
proved by the German mathematician Carl Friedrich Gauss (1777–1855).
Using the Fundamental Theorem of Algebra and the equivalence of zeros
and factors, you obtain the Linear Factorization Theorem.
Note that neither the Fundamental Theorem of Algebra nor the Linear
Factorization Theorem tells you how to find the zeros or factors of a polynomi-
al. Such theorems are called existence theorems. To find the zeros of a polyno-
mial function, you still must rely on other techniques.
Remember that the n zeros of a polynomial function can be real or complex,
and they may be repeated. Examples 1 and 2 illustrate several cases.
Linear Factorization Theorem (See the proof on page 332.)
If is a polynomial of degree where has precisely linear
factors
where are complex numbers.c
1
, c
2
, . . . , c
n
f
共
x
兲
a
n
共
x c
1
兲共
x c
2
兲
. . .
共
x c
n
兲
nfn
>
0,n,f
共
x
兲
The Fundamental Theorem of Algebra
If is a polynomial of degree where then has at least one zero
in the complex number system.
fn
>
0,n,f
共
x
兲
Example 1 Real Zeros of a Polynomial Function
Counting multiplicity, confirm that the second-degree polynomial function
has exactly two zeros: and
Solution
Repeated solution
The graph in Figure 3.38 touches the x-axis at
Now try Exercise 1.
x 3.
x 3 x 3 0
x
2
6x 9
共
x 3
兲
2
0
x 3.x 3
f
共
x
兲
x
2
6x 9
8−1
−1
5
f(x) = x
2
− 6x + 9
Figure 3.38
333371_0304.qxp 12/27/06 1:28 PM Page 291
292 Chapter 3 Polynomial and Rational Functions
Example 3 Finding the Zeros of a Polynomial Function
Write as the product of linear factors, and list
all the zeros of f.
Solution
The possible rational zeros are and The graph shown in Figure
3.40 indicates that 1 and are likely zeros, and that 1 is possibly a repeated zero
because it appears that the graph touches (but does not cross) the x-axis at this
point. Using synthetic division, you can determine that is a zero and 1 is a
repeated zero of f. So, you have
By factoring as
you obtain
which gives the following five zeros of f.
and
Note from the graph of f shown in Figure 3.40 that the real zeros are the only ones
that appear as x-intercepts.
Now try Exercise 27.
x 2ix 2i,x 2,x 1,x 1,
f
共
x
兲
共
x 1
兲共
x 1
兲共
x 2
兲共
x 2i
兲共
x 2i
兲
x
2
共
4
兲
共
x
冪
4
兲共
x
冪
4
兲
共
x 2i
兲共
x 2i
兲
x
2
4
共
x 1
兲共
x 1
兲共
x 2
兲共
x
2
4
兲
. f
共
x
兲
x
5
x
3
2x
2
12x 8
2
2
±
8.
±
1,
±
2,
±
4,
f
共
x
兲
x
5
x
3
2x
2
12x 8
Example 2 Real and Complex Zeros of a Polynomial Function
Confirm that the third-degree polynomial function
has exactly three zeros: and
Solution
Factor the polynomial completely as So, the zeros are
In the graph in Figure 3.39, only the real zero appears as an -intercept.
Now try Exercise 3.
Example 3 shows how to use the methods described in Sections 3.2 and 3.3
(the Rational Zero Test, synthetic division, and factoring) to find all the zeros of
a polynomial function, including complex zeros.
xx 0
x 2i. x 2i 0
x 2i x 2i 0
x 0
x
共
x 2i
兲共
x 2i
兲
0
x
共
x 2i
兲共
x 2i
兲
.
x 2i.x 0, x 2i,
f
共
x
兲
x
3
4x
−99
6
−6
f(x) = x
3
+ 4x
Figure 3.39
−33
16
−4
f(x) = x
5
+ x
3
+ 2x
2
− 12x + 8
Figure 3.40
You may want to remind students that a
graphing utility is helpful for determining
real zeros, which in turn are useful in
finding complex zeros.
333371_0304.qxp 12/27/06 1:28 PM Page 292
Conjugate Pairs
In Example 3, note that the two complex zeros are conjugates. That is, they are
of the forms and a bi.a bi
Be sure you see that this result is true only if the polynomial function has real
coefficients. For instance, the result applies to the function but not
to the function g
共
x
兲
x i.
f
共
x
兲
x
2
1,
Factoring a Polynomial
The Linear Factorization Theorem states that you can write any nth-degree
polynomial as the product of n linear factors.
However, this result includes the possibility that some of the values of are
complex. The following theorem states that even if you do not want to get
involved with “complex factors,” you can still write as the product of linear
and/or quadratic factors.
f
共
x
兲
c
i
f
共
x
兲
a
n
共
x c
1
兲共
x c
2
兲共
x c
3
兲
. . .
共
x c
n
兲
Section 3.4 The Fundamental Theorem of Algebra 293
Complex Zeros Occur in Conjugate Pairs
Let be a polynomial function that has real coefficients. If where
is a zero of the function, the conjugate is also a zero of the
function.
a bib 0,
a bi,f
共
x
兲
Example 4 Finding a Polynomial with Given Zeros
Find a fourth-degree polynomial function with real coefficients that has
and as zeros.
Solution
Because is a zero and the polynomial is stated to have real coefficients, you
know that the conjugate must also be a zero. So, from the Linear
Factorization Theorem, can be written as
For simplicity, let to obtain
Now try Exercise 39.
x
4
2x
3
10x
2
18x 9. f
共
x
兲
共
x
2
2x 1
兲共
x
2
9
兲
a 1
f
共
x
兲
a
共
x 1
兲共
x 1
兲共
x 3i
兲共
x 3i
兲
.
f
共
x
兲
3i
3i
3i
1,1,
Factors of a Polynomial (See the proof on page 332.)
Every polynomial of degree with real coefficients can be written as
the product of linear and quadratic factors with real coefficients, where the
quadratic factors have no real zeros.
n
>
0
333371_0304.qxp 12/27/06 1:28 PM Page 293
A quadratic factor with no real zeros is said to be prime or irreducible over
the reals. Be sure you see that this is not the same as being irreducible over the
rationals. For example, the quadratic
is irreducible over the reals (and therefore over the rationals). On the other hand,
the quadratic
is irreducible over the rationals, but reducible over the reals.
x
2
2
共
x
冪
2
兲共
x
冪
2
兲
x
2
1
共
x i
兲共
x i
兲
294 Chapter 3 Polynomial and Rational Functions
STUDY TIP
Recall that irrational and rational
numbers are subsets of the set
of real numbers, and the real
numbers are a subset of the set
of complex numbers.
Example 5 Factoring a Polynomial
Write the polynomial
a. as the product of factors that are irreducible over the rationals,
b. as the product of linear factors and quadratic factors that are irreducible over
the reals, and
c. in completely factored form.
Solution
a. Begin by factoring the polynomial into the product of two quadratic
polynomials.
Both of these factors are irreducible over the rationals.
b. By factoring over the reals, you have
where the quadratic factor is irreducible over the reals.
c. In completely factored form, you have
Now try Exercise 47.
In Example 5, notice from the completely factored form that the fourth-
degree polynomial has four zeros.
Throughout this chapter, the results and theorems have been stated in terms
of zeros of polynomial functions. Be sure you see that the same results could have
been stated in terms of solutions of polynomial equations. This is true because the
zeros of the polynomial function
are precisely the solutions of the polynomial equation
a
2
x
2
a
1
x a
0
0.
a
n
x
n
a
n 1
x
n 1
. . .
a
1
x a
0
. . .
a
2
x
2
f
共
x
兲
a
n
x
n
a
n 1
x
n 1
x
4
x
2
20
共
x
冪
5
兲共
x
冪
5
兲
共
x 2i
兲共
x 2i
兲
.
x
4
x
2
20
共
x
冪
5
兲共
x
冪
5
兲
共
x
2
4
兲
x
4
x
2
20
共
x
2
5
兲共
x
2
4
兲
f
共
x
兲
x
4
x
2
20
Activities
1. Write as a product of linear factors:
Answer:
2. Find a third-degree polynomial with
integer coefficients that has 2 and
as zeros.
Answer:
3. Write the polynomial
in
completely factored form. (Hint: One
factor is
Answer:
共
x 1
冪
2 i
兲共
x 1
冪
2 i
兲
共
x 1
冪
3
兲共
x 1
冪
3
兲
x
2
2x 2.
兲
f
共
x
兲
x
4
4x
3
5x
2
2x 6
x
3
8x
2
22x 20
3 i
共
x 2
兲共
x 2
兲共
x 2i
兲共
x 2i
兲
f
共
x
兲
x
4
16.
333371_0304.qxp 12/27/06 1:28 PM Page 294
In Example 6, if you were not told that is a zero of you
could still find all zeros of the function by using synthetic division to
find the real zeros and 3. Then, you could factor the polynomial as
Finally, by using the Quadratic Formula, you
could determine that the zeros are and x 2.x 3,x 1 3i,x 1 3i,
共
x 2
兲共
x 3
兲共
x
2
2x 10
兲
.
2
f,1 3i
Section 3.4 The Fundamental Theorem of Algebra
295
Algebraic Solution
Because complex zeros occur in conjugate pairs, you know that
is also a zero of f. This means that both
and
are factors of f. Multiplying these two factors produces
Using long division, you can divide into f to obtain
the following.
So, you have
and you can conclude that the zeros of are
and
Now try Exercise 53.
x 2.x 3,
x 1 3i,x 1 3i,f
共
x
2
2x 10
兲共
x 3
兲共
x 2
兲
f
共
x
兲
共
x
2
2x 10
兲共
x
2
x 6
兲
0
6x
2
12x 60
6x
2
12x 60
x
3
2x
2
10x
x
3
4x
2
2x
x
4
2x
3
10x
2
x
2
2x 10
)
x
4
3x
3
6x
2
2x 60
x
2
x 6
x
2
2x 10
x
2
2x 10.
共
x 1
兲
2
9i
2
关
x
共
1 3i
兲兴关
x
共
1 3i
兲兴
关共
x 1
兲
3i
兴关共
x 1
兲
3i
兴
x
共
1 3i
兲
x
共
1 3i
兲
1 3i
Graphical Solution
Because complex zeros always occur in conju-
gate pairs, you know that is also a zero of
f. Because the polynomial is a fourth-degree
polynomial, you know that there are at most two
other zeros of the function. Use a graphing
utility to graph
as shown in Figure 3.41.
Figure 3.41
You can see that and 3 appear to be
x-intercepts of the graph of the function. Use the
zero or root feature or the zoom and trace
features of the graphing utility to confirm that
and are x-intercepts of the graph.
So, you can conclude that the zeros of f are
and
x 2.
x 3,x 1 3i,x 1 3i,
x 3x 2
2
−55
60
−80
y = x
4
− 3x
3
+ 6x
2
+ 2x − 60
x = 3x = −2
y x
4
3x
3
6x
2
2x 60
1 3i
Example 6 Finding the Zeros of a Polynomial Function
Find all the zeros of
given that is a zero of f.1 3i
f
共
x
兲
x
4
3x
3
6x
2
2x 60
333371_0304.qxp 12/27/06 1:28 PM Page 295
296 Chapter 3 Polynomial and Rational Functions
In Exercises 1– 4, find all the zeros of the function.
1.
2.
3.
4.
Graphical and Analytical Analysis In Exercises 5–8, find all
the zeros of the function. Is there a relationship between the
number of real zeros and the number of x-intercepts of the
graph? Explain.
5. 6.
7. 8.
In Exercises 9–28, find all the zeros of the function and
write the polynomial as a product of linear factors. Use a
graphing utility to graph the function to verify your results
graphically. (If possible, use your graphing utility to verify
the complex zeros.)
9. 10.
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29–36, (a) find all zeros of the function,
(b) write the polynomial as a product of linear factors, and
(c) use your factorization to determine the x-intercepts of
the graph of the function. Use a graphing utility to verify
that the real zeros are the only x-intercepts.
29.
30.
31.
32.
33.
34.
35.
36. f
共
x
兲
x
4
8x
3
17x
2
8x 16
f
共
x
兲
x
4
25x
2
144
f
共
x
兲
x
3
10x
2
33x 34
f
共
x
兲
x
3
11x 150
f
共
x
兲
2x
3
5x
2
18x 45
f
共
x
兲
2x
3
3x
2
8x 12
f
共
x
兲
x
2
12x 34
f
共
x
兲
x
2
14x 46
h
共
x
兲
x
4
6x
3
10x
2
6x 9
g
共
x
兲
x
4
4x
3
8x
2
16x 16
f
共
s
兲
3s
3
4s
2
8s 8
f
共
x
兲
5x
3
9x
2
28x 6
f
共
x
兲
x
3
11x
2
39x 29
f
共
t
兲
t
3
3t
2
15t 125
f
共
x
兲
3x
3
2x
2
75x 50
f
共
x
兲
3x
3
5x
2
48x 80
f
共
x
兲
x
4
29x
2
100
f
共
x
兲
x
4
10x
2
9
h(x) x
2
4x 3
f
共
z
兲
z
2
z 56
f
共
y
兲
81y
4
625
f
共
x
兲
16x
4
81
f
共
x
兲
x
2
36f
共
x
兲
x
2
25
f
共
x
兲
x
2
6x 2f
共
x
兲
x
2
12x 26
g
共
x
兲
x
2
10x 23h
共
x
兲
x
2
4x 1
−66
−7
1
−33
−2
18
f
共
x
兲
x
4
3x
2
4f
共
x
兲
x
4
4x
2
4
−46
−10
20
−37
−13
2
4x 16 x 4
f
共
x
兲
x
3
4x
2
f
共
x
兲
x
3
4x
2
h
共
t
兲
共
t 3
兲共
t 2
兲共
t 3i
兲共
t 3i
兲
f
共
x
兲
共
x 9
兲共
x 4i
兲共
x 4i
兲
g
共
x)
共
x 2
兲共
x 4
兲
3
f
共
x
兲
x
2
共
x 3
兲
3.4 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
Fill in the blanks.
1. The _______ of _______ states that if is a polynomial function of degree then has at least one zero
in the complex number system.
2. The _______ states that if is a polynomial of degree then has precisely linear factors
where are complex numbers.
3. A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said
to be _______ over the _______ .
4. If is a complex zero of a polynomial with real coefficients, then so is its _______ .a bi
c
1
, c
2
,
. . .
, c
n
f
共
x
兲
a
n
共
x c
1
兲共
x c
2
兲
. . .
共
x c
n
兲
nfn,f
共
x
兲
fn
共
n
>
0
兲
,f
共
x
兲
333371_0304.qxp 12/27/06 1:28 PM Page 296
Section 3.4 The Fundamental Theorem of Algebra 297
In Exercises 37–42, find a polynomial function with real
coefficients that has the given zeros. (There are many
correct answers.)
37. 38.
39. 40.
41. 42.
In Exercises 43–46, the degree, the zeros, and a solution point
of a polynomial function f are given. Write f (a) in completely
factored form and (b) in expanded form.
Degree Zeros Solution Point
43. 4
44. 4
45. 3
46. 3
In Exercises 47–50, write the polynomial (a) as the product
of factors that are irreducible over the rationals, (b) as the
product of linear and quadratic factors that are irreducible
over the reals, and (c) in completely factored form.
47. 48.
49.
(Hint: One factor is )
50.
(Hint: One factor is )
In Exercises 51–58, use the given zero to find all the zeros of
the function.
Function Zero
51.
52.
53.
54.
55.
56.
57.
58.
Graphical Analysis In Exercises 59–62, (a) use the zero or
root feature of a graphing utility to approximate the zeros of
the function accurate to three decimal places and (b) find
the exact values of the remaining zeros.
59.
60.
61.
62.
63. Height A baseball is thrown upward from ground level
with an initial velocity of 48 feet per second, and its height
(in feet) is given by
where is the time (in seconds). You are told that the ball
reaches a height of 64 feet. Is this possible? Explain.
64. Profit The demand equation for a microwave is
where is the unit price (in dollars)
of the microwave and is the number of units produced and
sold. The cost equation for the microwave is
where is the total cost (in dollars)
and is the number of units produced. The total profit
obtained by producing and selling units is given by
You are working in the marketing
department of the company that produces this microwave,
and you are asked to determine a price that would yield a
profit of $9 million. Is this possible? Explain.
Synthesis
True or False? In Exercises 65 and 66, decide whether the
statement is true or false. Justify your answer.
65. It is possible for a third-degree polynomial function with
integer coefficients to have no real zeros.
66. If is a zero of the function
then must also be a zero of
67. Exploration Use a graphing utility to graph the function
for different values of Find values
of such that the zeros of satisfy the specified character-
istics. (Some parts have many correct answers.)
(a) Two real zeros, each of multiplicity 2
(b) Two real zeros and two complex zeros
68. Writing Compile a list of all the various techniques for
factoring a polynomial that have been covered so far in the
text. Give an example illustrating each technique, and write
a paragraph discussing when the use of each technique is
appropriate.
Skills Review
In Exercises 69–72, sketch the graph of the quadratic func-
tion. Identify the vertex and any intercepts. Use a graphing
utility to verify your results.
69.
70.
71.
72. f
共
x
兲
4x
2
2x 12
f
共
x
兲
6x
2
5x 6
f
共
x
兲
x
2
x 6
f
共
x
兲
x
2
7x 8
fk
k.f
共
x
兲
x
4
4x
2
k
f.x 4 3i
13x
2
265x 750f
共
x
兲
x
4
7x
3
x 4 3i
p
P R C xp C.
x
x
CC 80x 150,000,
x
pp 140 0.0001x,
t
h
共
t
兲
16t
2
48t, 0
≤
t
≤
3
h
f
共
x
兲
25x
3
55x
2
54x 18
h
共
x
兲
8x
3
14x
2
18x 9
f
共
x
兲
x
3
4x
2
14x 20
f
共
x
兲
x
4
3x
3
5x
2
21x 22
1
5
共
2
冪
2i
兲
f
共
x
兲
25x
3
55x
2
54x 18
1
2
共
1
冪
5i
兲
h
共
x
兲
8x
3
14x
2
18x 9
1 3if
共
x
兲
x
3
4x
2
14x 20
1
冪
3ih
共
x
兲
3x
3
4x
2
8x 8
3 ig
共
x
兲
4x
3
23x
2
34x 10
5 2ig
共
x
兲
x
3
7x
2
x 87
3if
共
x
兲
x
3
x
2
9x 9
5if
共
x
兲
2x
3
3x
2
50x 75
x
2
4.
f
共
x
兲
x
4
3x
3
x
2
12x 20
x
2
6.
f
共
x
兲
x
4
2x
3
3x
2
12x 18
f
共
x
兲
x
4
6x
2
27f
共
x
兲
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