Download 5.4Factor and Solve Polynomial Equations

5.4
Before
Now
Why?
Key Vocabulary
• factored completely
• factor by grouping
• quadratic form
Factor and Solve
Polynomial Equations
You factored and solved quadratic equations.
You will factor and solve other polynomial equations.
So you can find dimensions of archaeological ruins, as in Ex. 58.
In Chapter 4, you learned how to factor the following types of quadratic
expressions.
Type
Example
General trinomial
2x2 2 3x 2 20 5 (2x 1 5)(x 2 4)
Perfect square trinomial
x2 1 8x 1 16 5 (x 1 4) 2
Difference of two squares
9x2 2 1 5 (3x 1 1)(3x 2 1)
Common monomial factor
8x2 1 20x 5 4x(2x 1 5)
You can also factor polynomials with degree greater than 2. Some of these
polynomials can be factored completely using techniques learned in Chapter 4.
For Your Notebook
KEY CONCEPT
Factoring Polynomials
Definition
Examples
A factorable polynomial with integer
coefficients is factored completely
if it is written as a product of
unfactorable polynomials with
integer coefficients.
2(x 1 1)(x 2 4) and 5x2 (x2 2 3) are
factored completely.
EXAMPLE 1
3x(x2 2 4) is not factored completely
because x2 2 4 can be factored as
(x 1 2)(x 2 2).
Find a common monomial factor
Factor the polynomial completely.
a. x 3 1 2x2 2 15x 5 x(x 2 1 2x 2 15)
Factor common monomial.
5 x(x 1 5)(x 2 3)
b. 2y 5 2 18y 3 5 2y 3 (y 2 2 9)
Factor trinomial.
Factor common monomial.
3
5 2y (y 1 3)(y 2 3)
Difference of two squares
c. 4z 4 2 16z3 1 16z2 5 4z2 (z2 2 4z 1 4)
2
5 4z (z 2 2)
2
Factor common monomial.
Perfect square trinomial
5.4 Factor and Solve Polynomial Equations
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FACTORING PATTERNS In part (b) of Example 1, the special factoring pattern
for the difference of two squares is used to factor the expression completely.
There are also factoring patterns that you can use to factor the sum or
difference of two cubes.
For Your Notebook
KEY CONCEPT
Special Factoring Patterns
Sum of Two Cubes
Example
a3 1 b3 5 (a 1 b)(a2 2 ab 1 b2)
8x3 1 27 5 (2x) 3 1 33
5 (2x 1 3)(4x2 2 6x 1 9)
Difference of Two Cubes
Example
a 2 b 5 (a 2 b)(a 1 ab 1 b
3
3
2
)
2
64x 3 2 1 5 (4x) 3 2 13
5 (4x 2 1)(16x2 1 4x 1 1)
EXAMPLE 2
Factor the sum or difference of two cubes
Factor the polynomial completely.
a. x 3 1 64 5 x 3 1 43
Sum of two cubes
2
5 (x 1 4)(x 2 4x 1 16)
5
b. 16z 2 250z2 5 2z2 (8z3 2 125)
Factor common monomial.
5 2z2 F(2z) 3 2 53 G
2
Difference of two cubes
2
5 2z (2z 2 5)(4z 1 10z 1 25)
✓
GUIDED PRACTICE
for Examples 1 and 2
Factor the polynomial completely.
1. x 3 2 7x2 1 10x
2. 3y 5 2 75y 3
3. 16b5 1 686b2
4. w 3 2 27
FACTORING BY GROUPING For some polynomials, you can factor by grouping
pairs of terms that have a common monomial factor. T he pattern for factori ng by
grouping is shown below.
ra 1 rb 1 sa 1 sb 5 r(a 1 b) 1 s(a 1 b)
5 (r 1 s)(a 1 b)
EXAMPLE 3
AVOID ERRORS
An expression is not
factored completely
until all factors, such
as x2 2 16, cannot be
factored further.
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Factor by grouping
Factor the polynomial x 3 2 3x 2 2 16x 1 48 completely.
x 3 2 3x2 2 16x 1 48 5 x 2 (x 2 3) 2 16(x 2 3)
2
Factor by grouping.
5 (x 2 16)(x 2 3)
Distributive property
5 (x 1 4)(x 2 4)(x 2 3)
Difference of two squares
Chapter 5 Polynomials and Polynomial Functions
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QUADRATIC FORM An expression of the form au2 1 bu 1 c, where u is any
expression in x, is said to be in quadratic form. The factoring techniques you
studied in Chapter 4 can sometimes be used to factor such expressions.
EXAMPLE 4
IDENTIFY
QUADRATIC FORM
The expression
16x 4 2 81 is in
quadratic form because
it can be written as
u2 2 81 where u 5 4x2.
Factor polynomials in quadratic form
Factor completely: (a) 16x 4 2 81 and (b) 2p8 1 10p5 1 12p2 .
a. 16x4 2 81 5 (4x 2)2 2 92
Write as difference of two squares.
5 (4x2 1 9)(4x2 2 9)
Difference of two squares
2
8
5 (4x 1 9)(2x 1 3)(2x 2 3)
Difference of two squares
5
Factor common monomial.
2
2
6
3
b. 2p 1 10p 1 12p 5 2p (p 1 5p 1 6)
5 2p2 (p3 1 3)(p3 1 2)
✓
GUIDED PRACTICE
Factor trinomial in quadratic form.
for Examples 3 and 4
Factor the polynomial completely.
5. x 3 1 7x2 2 9x 2 63
6. 16g 4 2 625
7. 4t 6 2 20t 4 1 24t 2
SOLVING POLYNOMIAL EQUATIONS In Chapter 4, you learned how to use the zero
product property to solve factorable quadratic equations. You can extend this
technique to solve some higher-degree polynomial equations.
★
EXAMPLE 5
Standardized Test Practice
What are the real-number solutions of the equation 3x 5 1 15x 5 18x 3 ?
A 0, 1, 3, 5
B 21, 0, 1
}
}
}
D 2Ï 5 , 21, 0, 1, Ï5
C 0, 1, Ï 5
Solution
3x5 1 15x 5 18x3
5
Write original equation.
3
3x 2 18x 1 15x 5 0
AVOID ERRORS
4
Do not divide each side
of an equation by a
variable or a variable
expression, such as 3x.
Doing so will result in
the loss of solutions.
✓
Write in standard form.
2
3x(x 2 6x 1 5) 5 0
Factor common monomial.
3x(x2 2 1)(x2 2 5) 5 0
Factor trinomial.
2
3x(x 1 1)(x 2 1)(x 2 5) 5 0
Difference of two squares
}
}
x 5 0, x 5 21, x 5 1, x 5 Ï5 , or x 5 2Ï 5
Zero product property
c The correct answer is D. A B C D
GUIDED PRACTICE
for Example 5
Find the real-number solutions of the equation.
8. 4x5 2 40x 3 1 36x 5 0
9. 2x5 1 24x 5 14x 3
10. 227x 3 1 15x 2 5 26x4
5.4 Factor and Solve Polynomial Equations
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EXAMPLE 6
Solve a polynomial equation
CITY PARK You are designing a marble basin
FT
that will hold a fountain for a city park. The
basin’s sides and bottom should be 1 foot thick.
Its outer length should be twice its outer width
and outer height.
X
What should the outer dimensions of the basin
be if it is to hold 36 cubic feet of water?
X
X
Solution
ANOTHER WAY
For alternative methods
to solving the problem
in Example 6, turn
to page 360 for the
Problem Solving
Workshop.
Volume
(cubic feet)
5
36
5
Interior length
(feet)
3
2
(feet)
p
(2x 2 2)
36 5 (2x 2 2)(x 2 2)(x 2 1)
Interior width
p
(x 2 2)
Write in standard form.
0 5 2x2 (x 2 4) 1 10(x 2 4)
Factor by grouping.
0 5 (2x 1 10)(x 2 4)
Interior height
p
(feet)
(x 2 1)
Write equation.
0 5 2x 2 8x 1 10x 2 40
2
p
Distributive property
c The only real solution is x 5 4. The basin is 8 ft long, 4 ft wide, and 4 ft high.
✓
GUIDED PRACTICE
for Example 6
11. WHAT IF? In Example 6, what should the basin’s dimensions be if it is to hold
128 cubic feet of water and have outer length 6x, width 3x, and height x?
5.4
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 7, 23, and 61
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 9, 41, 63, and 64
SKILL PRACTICE
1. VOCABULARY The expression 8x6 1 10x 3 2 3 is in ? form because it can be
written as 2u2 1 5u 2 3 where u 5 2x 3.
2. ★ WRITING What condition must the factorization of a polynomial satisfy in
order for the polynomial to be factored completely?
EXAMPLE 1
on p. 353
for Exs. 3–9
MONOMIAL FACTORS Factor the polynomial completely.
3. 14x 2 2 21x
4. 30b3 2 54b2
5. c 3 1 9c 2 1 18c
6. z3 2 6z2 2 72z
7. 3y 5 2 48y 3
8. 54m5 1 18m 4 1 9m3
9. ★ MULTIPLE CHOICE What is the complete factorization of 2x 7 2 32x 3 ?
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A 2x3 (x 1 2)(x 2 2)(x2 1 4)
B 2x3 (x2 1 2)(x2 2 2)
C 2x3 (x2 1 4)2
D 2x 3 (x 1 2)2 (x 2 2)2
Chapter 5 Polynomials and Polynomial Functions
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EXAMPLE 2
SUM OR DIFFERENCE OF CUBES Factor the polynomial completely.
on p. 354
for Exs. 10–17
10. x 3 1 8
11. y 3 2 64
12. 27m3 1 1
13. 125n3 1 216
14. 27a 3 2 1000
15. 8c 3 1 343
16. 192w 3 2 3
17. 25z3 1 320
EXAMPLE 3
FACTORING BY GROUPING Factor the polynomial completely.
on p. 354
for Exs. 18–23
18. x 3 1 x2 1 x 1 1
19. y 3 2 7y 2 1 4y 2 28
20. n3 1 5n2 2 9n 2 45
21. 3m3 2 m2 1 9m 2 3
22. 25s 3 2 100s 2 2 s 1 4
23. 4c 3 1 8c 2 2 9c 2 18
EXAMPLE 4
QUADRATIC FORM Factor the polynomial completely.
on p. 355
for Exs. 24–29
24. x4 2 25
25. a 4 1 7a2 1 6
26. 3s4 2 s 2 2 24
27. 32z5 2 2z
28. 36m6 1 12m4 1 m2
29. 15x5 2 72x 3 2 108x
EXAMPLE 5
ERROR ANALYSIS Describe and correct the error in finding all real-number
on p. 355
for Exs. 30–41
solutions.
30.
31.
8x3 2 27 5 0
2
(2x 1 3)(4x 1 6x 1 9) 5 0
3
x 5 2}
2
3x3 2 48x 5 0
3x(x2 2 16) 5 0
x2 2 16 5 0
x 5 24 or x 5 4
SOLVING EQUATIONS Find the real-number solutions of the equation.
32. y 3 2 5y 2 5 0
33. 18s 3 5 50s
34. g 3 1 3g 2 2 g 2 3 5 0
35. m3 1 6m2 2 4m 2 24 5 0
36. 4w4 1 40w 2 2 44 5 0
37. 4z5 5 84z3
38. 5b3 1 15b2 1 12b 5 236
39. x6 2 4x4 2 9x2 1 36 5 0
40. 48p5 5 27p3
41. ★ MULTIPLE CHOICE What are the real-number solutions of the equation
3x4 2 27x2 1 9x 5 x3 ?
A 21, 0, 3
1, 3
C 23, 0, }
B 23, 0, 3
3
1 , 0, 3
D 23, 2}
3
CHOOSING A METHOD Factor the polynomial completely using any method.
42. 16x 3 2 44x 2 2 42x
43. n 4 2 4n2 2 60
44. 24b4 2 500b
45. 36a 3 2 15a2 1 84a 2 35
46. 18c4 1 57c 3 2 10c 2
47. 2d 4 2 13d2 2 45
48. 32x5 2 108x 2
49. 8y 6 2 38y4 2 10y 2
50. z5 2 3z 4 2 16z 1 48
GEOMETRY Find the possible value(s) of x.
51. Area 5 48
52. Volume 5 40
53. Volume 5 125π
x24
2x 2 5
x 14
2x
3x 1 2
3x
x21
CHOOSING A METHOD Factor the polynomial completely using any method.
54. x 3y 6 2 27
55. 7ac 2 1 bc 2 2 7ad2 2 bd2
56. x2n 2 2xn 1 1
57. CHALLENGE Factor a5b2 2 a2b4 1 2a 4b 2 2ab3 1 a 3 2 b2 completely.
5.4 Factor and Solve Polynomial Equations
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PROBLEM SOLVING
EXAMPLE 6
on p. 356
for Exs. 58–63
58. ARCHAEOLOGY At the ruins of Caesarea,
archaeologists discovered a huge hydraulic
concrete block with a volume of 945 cubic
meters. The block’s dimensions are x meters
high by 12x 2 15 meters long by 12x 2 21 meters
wide. What is the height of the block?
LEBANON
SYRIA
Caesarea
ISRAEL
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
EGYPT
JORDAN
59. CHOCOLATE MOLD You are designing a chocolate mold shaped like a hollow
rectangular prism for a candy manufacturer. The mold must have a thickness
of 1 centimeter in all dimensions. The mold’s outer dimensions should also
be in the ratio 1: 3 : 6. What should the outer dimensions of the mold be if it is
to hold 112 cubic centimeters of chocolate?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
60. MULTI-STEP PROBLEM A production crew is assembling a three-level
platform inside a stadium for a performance. The platform has the
dimensions shown in the diagrams, and has a total volume of 1250 cubic feet.
2x 4x 6x
x
4x
x
6x
x
8x
a. Write Expressions What is the volume, in terms of x, of each of the three
levels of the platform?
b. Write an Equation Use what you know about the total volume to write an
equation involving x.
c. Solve Solve the equation from part (b). Use your solution to calculate the
dimensions of each of the three levels of the platform.
61. SCULPTURE Suppose you have 250 cubic inches of clay with which to make
a sculpture shaped as a rectangular prism. You want the height and width
each to be 5 inches less than the length. What should the dimensions of the
prism be?
62. MANUFACTURING A manufacturer wants to build a rectangular
stainless steel tank with a holding capacity of 670 gallons, or
about 89.58 cubic feet. The tank’s walls will be one half inch
thick, and about 6.42 cubic feet of steel will be used for the tank.
The manufacturer wants the outer dimensions of the tank to be
related as follows:
x18
• The width should be 2 feet less than the length.
• The height should be 8 feet more than the length.
What should the outer dimensions of the tank be?
358
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5 WORKED-OUT SOLUTIONS
Chapter 5 Polynomials
on p. WS1 and Polynomial Functions
★
x
x22
5 STANDARDIZED
TEST PRACTICE
10/25/05 10:26:42 AM
63. ★ SHORT RESPONSE A platform shaped like a rectangular prism has
dimensions x 2 2 feet by 3 2 2x feet by 3x 1 4 feet. Explain why the volume
7 cubic feet.
of the platform cannot be }
3
64. ★ EXTENDED RESPONSE In 2000 B.C., the Babylonians solved polynomial
equations using tables of values. One such table gave values of y 3 1 y 2.
To be able to use this table, the Babylonians sometimes had to manipulate
the equation, as shown below.
ax3 1 bx 2 5 c
a3x 3
b
2 2
ax
b
Original equation
2
ac
b
1}
5}
}
2
3
3
ax
1}
b 2
3
2
ax 5 a c
1 1}
2 }3
2
b
b
a2
b
Multiply each side by }
.
3
Rewrite cubes and squares.
2
a c in the y 3 1 y 2 column of the table. Because the
They then found }
3
b
by
a
ax , they could conclude that x 5
corresponding y-value was y 5 }
}.
b
a. Calculate y 3 1 y 2 for y 5 1, 2, 3, . . . , 10. Record the values in a table.
b. Use your table and the method described above to solve x 3 1 2x2 5 96.
c. Use your table and the method described above to solve 3x 3 1 2x2 5 512.
d. How can you modify the method described above for equations of the
form ax4 1 bx 3 5 c?
65. CHALLENGE Use the diagram to complete parts (a)–(c).
III
II
a. Explain why a 3 2 b3 is equal to the sum of the volumes of
solid I, solid II, and solid III.
a
b
b. Write an algebraic expression for the volume of each of the
three solids. Leave your expressions in factored form.
b
b
I
c. Use the results from parts (a) and (b) to derive the
factoring pattern for a3 2 b3 given on page 354.
a
a
MIXED REVIEW
Graph the function.
66. f(x) 5 22x 2 3 1 5 (p. 123)
1 x 2 1 4x 1 5 (p. 236)
67. y 5 }
2
68. y 5 3(x 1 4)2 1 7 (p. 245)
69. f(x) 5 x 3 2 2x 2 5 (p. 337)
Graph the inequality in a coordinate plane. (p. 132)
PREVIEW
Prepare for
Lesson 5.5
in Exs. 76–79.
70. y ≤ 2x 2 3
71. y > 25 2 x
72. y < 0.5x 1 5
73. 4x 1 12y ≤ 4
74. 9x 2 9y ≥ 27
2x 1 5y > 5
75. }
}
5
2
Use synthetic substitution to evaluate the polynomial function for the given
value of x. (p. 337)
76. f(x) 5 5x4 2 3x 3 1 4x 2 2 x 1 10; x 5 2
77. f(x) 5 23x5 1 x 3 2 6x 2 1 2x 1 4; x 5 23
78. f(x) 5 5x5 2 4x 3 1 12x 2 1 20; x 5 22
79. f(x) 5 26x4 1 9x 2 15; x 5 4
EXTRA PRACTICE for Lesson 5.4, p. 1014
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5.4 ONLINE
Factor andQUIZ
Solve Polynomial
Equations
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Using
ALTERNATIVE METHODS
LESSON 5.4
Another Way to Solve Example 6, page 356
MULTIPLE REPRESENTATIONS In Example 6 on page 356, you solved a polynomial
equation by factoring. You can also solve a polynomial equation using a table or a
graph.
PROBLEM
CITY PARK You are designing a marble
FT
basin that will hold a fountain for a city
park. The basin’s sides and bottom should
be 1 foot thick. Its outer length should be
twice its outer width and outer height.
X
X
What should the outer dimensions of
the basin be if it is to hold 36 cubic feet
of water?
METHOD 1
X
Using a Table One alternative approach is to write a function for the volume of
the basin and make a table of values for the function. Using the table, you can
find the value of x that makes the volume of the basin 36 cubic feet.
STEP 1 Write the function. From the diagram, you can see that the volume y of
water the basin can hold is given by this function:
y 5 (2x 2 2)(x 2 2)(x 2 1)
STEP 2 Make a table of values for the
STEP 3 Identify the value of x for
function. Use only positive
values of x because the basin’s
dimensions must be positive.
X
1
2
3
4
5
Y1=96
Y1
which y 5 36. The table
shows that y 5 36 when
x 5 4.
X
1
2
3
4
5
Y1=96
0
0
8
36
96
Y1
0
0
8
36
96
c The volume of the basin is 36 cubic feet when x is 4 feet. So, the outer
dimensions of the basin should be as follows:
Length 5 2x 5 8 feet
Width 5 x 5 4 feet
Height 5 x 5 4 feet
360
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Chapter 5 Polynomials and Polynomial Functions
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METHOD 2
Using a Graph Another approach is to make a graph. You can use the graph to
find the value of x that makes the volume of the basin 36 cubic feet.
STEP 1 Write the function. From the diagram, you can see that the volume y of
water the basin can hold is given by this function:
y 5 (2x 2 2)(x 2 2)(x 2 1)
STEP 2 Graph the equations y 5 36
STEP 3 Identify the coordinates of
and y 5 (x 2 1)(2x 2 2)(x 2 2).
Choose a viewing window
that shows the intersection of
the graphs.
the intersection point. On a
graphing calculator, you can
use the intersect feature. The
intersection point is (4, 36).
Intersection
X=4
Y=36
c The volume of the basin is 36 cubic feet when x is 4 feet. So, the outer
dimensions of the basin should be as follows:
Length 5 2x 5 8 feet
Width 5 x 5 4 feet
Height 5 x 5 4 feet
P R AC T I C E
SOLVING EQUATIONS Solve the polynomial
equation using a table or using a graph.
1. x 3 1 4x2 2 8x 5 96
2. x3 2 9x2 2 14x 1 7 5 233
3. 2x3 2 11x2 1 3x 1 5 5 59
4. x4 1 x3 2 15x2 2 8x 1 6 5 245
5. 2x4 1 2x3 1 6x2 1 17x 2 4 5 32
6. 23x4 1 4x 3 1 8x2 1 4x 2 11 5 13
7. 4x4 2 16x 3 1 29x2 2 95x 5 2150
8. WHAT IF? In the problem on page 360, suppose
the basin is to hold 200 cubic feet of water. Find
the outer dimensions of the basin using a table
and using a graph.
9. PACKAGING A factory needs a box that has a
volume of 1728 cubic inches. The width should
be 4 inches less than the height, and the length
should be 6 inches greater than the height.
Find the dimensions of the box using a table
and using a graph.
10. AGRICULTURE From 1970 to 2002, the average
yearly pineapple consumption P (in pounds)
per person in the United States can be modeled
by the function
P(x) 5 0.0000984x4 2 0.00712x3 1 0.162x2 2
1.11x 1 12.3
where x is the number of years since 1970. In
what year was the pineapple consumption
about 9.97 pounds per person? Solve the
problem using a table and a graph.
Using Alternative Methods
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