5.4 factor and solve polynomial equations · 5.4 factor and solve polynomial equations 353...

5.4 Factor and Solve Polynomial Equations 353

5.4 Factor and Solve Polynomial Equations

Before You factored and solved quadratic equations.

Now You will factor and solve other polynomial equations.

Why? So you can find dimensions of archaeological ruins, as in Ex. 58.

Key Vocabulary • factored completely• factor by grouping• quadratic form

In Chapter 4, you learned how to factor the following types of quadratic expressions.

You can also factor polynomials with degree greater than 2. Some of these polynomials can be factored completely using techniques learned in Chapter 4.

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)

E X A M P L E 1 Find a common monomial factor

Factor the polynomial completely.

a. x3 1 2x2 2 15x 5 x(x2 1 2x 2 15) Factor common monomial.

5 x(x 1 5)(x 2 3) Factor trinomial.

b. 2y5 2 18y3 5 2y3(y2 2 9) Factor common monomial.

5 2y3(y 1 3)(y 2 3) Difference of two squares

c. 4z4 2 16z3 1 16z2 5 4z2(z2 2 4z 1 4) Factor common monomial.

5 4z2(z 2 2)2 Perfect square trinomial

KEY CONCEPT For Your Notebook

Factoring Polynomials

Defi nition 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.

3x(x2 2 4) is not factored completely because x2 2 4 can be factored as (x 1 2)(x 2 2).

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354 Chapter 5 Polynomials and Polynomial Functions

E X A M P L E 2 Factor the sum or difference of two cubes


a. x3 1 64 5 x3 1 43 Sum of two cubes

5 (x 1 4)(x2 2 4x 1 16)

b. 16z5 2 250z2 5 2z2(8z3 2 125) Factor common monomial.

5 2z2 F (2z)3 2 53 G Difference of two cubes

5 2z2(2z 2 5)(4z2 1 10z 1 25)

✓ GUIDED PRACTICE for Examples 1 and 2


1. x3 2 7x2 1 10x 2. 3y5 2 75y3 3. 16b5 1 686b2 4. w3 2 27

E X A M P L E 3 Factor by grouping

Factor the polynomial x3 2 3x2 2 16x 1 48 completely.

x3 2 3x2 2 16x 1 48 5 x2(x 2 3) 2 16(x 2 3) Factor by grouping.

5 (x2 2 16)(x 2 3) Distributive property

5 (x 1 4)(x 2 4)(x 2 3) Difference of two squares

FACTORING BY GROUPING For some polynomials, yopairs of terms that have a common monomial factor. ttern for factori y 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)

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.

KEY CONCEPT For Your Notebook

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

a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2) 64x3 2 1 5 (4x)3 2 13

5 (4x 2 1)(16x2 1 4x 1 1)

AVOID ERRORS

An expression is not factored completely until all factors, such as x2 2 16, cannot be factored further.


u ctor by groupi T

fng b

can a ng he pa

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E X A M P L E 4 Factor polynomials in quadratic form

Factor completely: (a) 16x4 2 81 and (b) 2p8 1 10p5 1 12p2.

a. 16x4 2 81 5 (4x2)2 2 92 Write as difference of two squares.

5 (4x2 1 9)(4x2 2 9) Difference of two squares

5 (4x2 1 9)(2x 1 3)(2x 2 3) Difference of two squares

b. 2p8 1 10p5 1 12p2 5 2p2(p6 1 5p3 1 6) Factor common monomial.

5 2p2(p3 1 3)(p3 1 2) Factor trinomial in quadratic form.

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.

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.

✓ GUIDED PRACTICE for Example 5

Find the real-number solutions of the equation.

8. 4x5 2 40x3 1 36x 5 0 9. 2x5 1 24x 5 14x3 10. 227x3 1 15x2 5 26x4

✓ GUIDED PRACTICE for Examples 3 and 4


5. x3 1 7x2 2 9x 2 63 6. 16g 4 2 625 7. 4t 6 2 20t 4 1 24t 2

IDENTIFY QUADRATIC FORM

The expression 16x4 2 81 is in quadratic form because it can be written as u2 2 81 where u 5 4x2.

★

Solution

3x5 1 15x 5 18x3 Write original equation.

3x5 2 18x3 1 15x 5 0 Write in standard form.

3x(x4 2 6x2 1 5) 5 0 Factor common monomial.

3x(x2 2 1)(x2 2 5) 5 0 Factor trinomial.

3x(x 1 1)(x 2 1)(x2 2 5) 5 0 Difference of two squares

x 5 0, x 5 21, x 5 1, x 5 Ï}

5 , or x 5 2 Ï}

5 Zero product property

c The correct answer is D. A B C D

E X A M P L E 5 Standardized Test Practice

What are the real-number solutions of the equation 3x5 1 15x 5 18x3?

A 0, 1, 3, 5 B 21, 0, 1

C 0, 1, Ï}

5 D 2 Ï}

5 , 21, 0, 1, Ï}

5

AVOID ERRORS

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.


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E X A M P L E 6 Solve a polynomial equation

CITY PARK You are designing a marble 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.

What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?

Solution

Volume (cubic feet)

5

Interior length (feet)

p

Interior width (feet)

p

Interior height (feet)

36 5 (2x 2 2) p (x 2 2) p (x 2 1)

36 5 (2x 2 2)(x 2 2)(x 2 1) Write equation.

0 5 2x3 2 8x2 1 10x 2 40 Write in standard form.

0 5 2x2(x 2 4) 1 10(x 2 4) Factor by grouping.

0 5 (2x2 1 10)(x 2 4) 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?

1. VOCABULARY The expression 8x6 1 10x3 2 3 is in ? form because it can be written as 2u2 1 5u 2 3 where u 5 2x3.

2. ★ WRITING What condition must the factorization of a polynomial satisfy in order for the polynomial to be factored completely?

MONOMIAL FACTORS Factor the polynomial completely.

3. 14x2 2 21x 4. 30b3 2 54b2 5. c3 1 9c2 1 18c

6. z3 2 6z2 2 72z 7. 3y5 2 48y3 8. 54m5 1 18m4 1 9m3

9. ★ MULTIPLE CHOICE What is the complete factorization of 2x7 2 32x3?

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 2x3(x 1 2)2(x 2 2)2

EXAMPLE 1

on p. 353 for Exs. 3–9

5.4 EXERCISES

ANOTHER WAY

For alternative methods to solving the problem in Example 6, turn to page 360 for the Problem Solving Workshop.

HOMEWORKKEY

5 WORKED-OUT SOLUTIONSon p. WS1 for Exs. 7, 23, and 61

★ 5 STANDARDIZED TEST PRACTICEExs. 2, 9, 41, 63, and 64

SKILL PRACTICE


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SUM OR DIFFERENCE OF CUBES Factor the polynomial completely.

10. x3 1 8 11. y3 2 64 12. 27m3 1 1 13. 125n3 1 216

14. 27a3 2 1000 15. 8c3 1 343 16. 192w3 2 3 17. 25z3 1 320

FACTORING BY GROUPING Factor the polynomial completely.

18. x3 1 x2 1 x 1 1 19. y3 2 7y2 1 4y 2 28 20. n3 1 5n2 2 9n 2 45

21. 3m3 2 m2 1 9m 2 3 22. 25s3 2 100s2 2 s 1 4 23. 4c3 1 8c2 2 9c 2 18

QUADRATIC FORM Factor the polynomial completely.

24. x4 2 25 25. a4 1 7a2 1 6 26. 3s4 2 s2 2 24

27. 32z5 2 2z 28. 36m6 1 12m4 1 m2 29. 15x5 2 72x3 2 108x

ERROR ANALYSIS Describe and correct the error in finding all real-number solutions.

30. 8x3 2 27 5 0 (2x 1 3)(4x2 1 6x 1 9) 5 0

x 5 2 3 } 2

31. 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. y3 2 5y2 5 0 33. 18s3 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 40w2 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 B 23, 0, 3 C 23, 0, 1 } 3

, 3 D 23, 2 1 } 3

, 0, 3

CHOOSING A METHOD Factor the polynomial completely using any method.

42. 16x3 2 44x2 2 42x 43. n4 2 4n2 2 60 44. 24b4 2 500b

45. 36a3 2 15a2 1 84a 2 35 46. 18c4 1 57c3 2 10c2 47. 2d4 2 13d2 2 45

48. 32x5 2 108x2 49. 8y6 2 38y4 2 10y2 50. z5 2 3z4 2 16z 1 48

GEOMETRY Find the possible value(s) of x.

51. Area 5 48 52. Volume 5 40 53. Volume 5 125π

3x 1 2

x 1 4

x 2 4

2x

x 2 1

2x 2 5

3x

CHOOSING A METHOD Factor the polynomial completely using any method.

54. x3y6 2 27 55. 7ac2 1 bc2 2 7ad2 2 bd2 56. x2n 2 2xn 1 1

57. CHALLENGE Factor a5b2 2 a2b4 1 2a4b 2 2ab3 1 a3 2 b2 completely.

EXAMPLE 2

on p. 354 for Exs. 10–17

EXAMPLE 3

on p. 354 for Exs. 18–23

EXAMPLE 4

on p. 355 for Exs. 24–29

EXAMPLE 5

on p. 355 for Exs. 30–41


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PROBLEM SOLVING

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?

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?

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.

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:

• 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?

5 WORKED-OUT SOLUTIONSon p. WS1

EXAMPLE 6

on p. 356 for Exs. 58–63

x 1 8

x 2 2x

ISRAEL

Caesarea

EGYPT JORDAN

SYRIALEBANON

6x4x2x

8x

6x

4x

xx

x

★ 5 STANDARDIZED TEST PRACTICE


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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

of the platform cannot be 7 } 3

cubic feet.

64. ★ EXTENDED RESPONSE In 2000 B.C., the Babylonians solved polynomial equations using tables of values. One such table gave values of y3 1 y2. To be able to use this table, the Babylonians sometimes had to manipulate the equation, as shown below.

ax3 1 bx2 5 c Original equation

a3x 3

} b3

1 a2x2

} b2

5 a2c }

b3 Multiply each side by a

2 }

b3 .

1 ax } b

2 3 1 1 ax }

b 2

2 5 a

2c } b3

Rewrite cubes and squares.

They then found a2c }

b3 in the y3 1 y2 column of the table. Because the

corresponding y-value was y 5 ax } b

, they could conclude that x 5 by

} a

.

a. Calculate y3 1 y2 for y 5 1, 2, 3, . . . , 10. Record the values in a table.

b. Use your table and the method described above to solve x3 1 2x2 5 96.

c. Use your table and the method described above to solve 3x3 1 2x2 5 512.

d. How can you modify the method described above for equations of the form ax4 1 bx3 5 c?

65. CHALLENGE Use the diagram to complete parts (a)–(c).

a. Explain why a3 2 b3 is equal to the sum of the volumes of solid I, solid II, and solid III.

b. Write an algebraic expression for the volume of each of the three solids. Leave your expressions in factored form.

c. Use the results from parts (a) and (b) to derive the factoring pattern for a3 2 b3 given on page 354.

EXTRA PRACTICE for Lesson 5.4, p. 1014 ONLINE QUIZ at classzone.com

a

bb

ba

a

I

II III

MIXED REVIEW

Graph the function.

66. f(x) 5 22x 2 3 1 5 (p. 123) 67. y 5 1 } 2

x2 1 4x 1 5 (p. 236)

68. y 5 3(x 1 4)2 1 7 (p. 245) 69. f(x) 5 x3 2 2x 2 5 (p. 337)

Graph the inequality in a coordinate plane. (p. 132)

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 75. 2 } 5

x 1 5 } 2

y > 5

Use synthetic substitution to evaluate the polynomial function for the given value of x. (p. 337)

76. f(x) 5 5x4 2 3x3 1 4x2 2 x 1 10; x 5 2 77. f(x) 5 23x5 1 x3 2 6x2 1 2x 1 4; x 5 23

78. f(x) 5 5x5 2 4x3 1 12x2 1 20; x 5 22 79. f(x) 5 26x4 1 9x 2 15; x 5 4

PREVIEW

Prepare for Lesson 5.5 in Exs. 76–79.


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ALTERNATIVE METHODSALTERNATIVE METHODSUsingUsing

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 function. Use only positive values of x because the basin’s dimensions must be positive.

Y10083696

X12345Y1=96

STEP 3 Identify the value of x for which y 5 36. The table shows that y 5 36 when x 5 4.

Y10083696

X12345Y1=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

LESSON 5.4Another 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.

CITY PARK You are designing a marble 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.

What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?

PRO B L E M

M E T H O D 1


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Using Alternative Methods 361

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 and y 5 (x 2 1)(2x 2 2)(x 2 2). Choose a viewing windowthat shows the intersection of the graphs.

STEP 3 Identify the coordinates of the intersection point. On a graphing calculator, you can use the intersect feature. The intersection point is (4, 36).

IntersectionX=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

SOLVING EQUATIONS Solve the polynomial equation using a table or using a graph.

1. x3 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 4x3 1 8x2 1 4x 2 11 5 13

7. 4x4 2 16x3 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.

PR AC T I C E

M E T H O D 2


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5.4 factor and solve polynomial equations · 5.4 factor and solve polynomial equations 353...

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