TEKS 5.4 2A.1.A, 2A.2.A; P..A, P..B 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 Vocaulary factored completely factor y grouping quadratic form In Chapter 4, you learned how to factor the following types of quadratic expressions. Type Example General trinomial 2x 2 2 x 2 20 5 (2x 1 5)(x 2 4) Perfect square trinomial x 2 1 8x 1 16 5 (x 1 4) 2 Difference of two squares 9x 2 2 1 5 (x 1 1)(x 2 1) Common monomial factor 8x 2 1 20x 5 4x(2x 1 5) You can also factor polynomials with degree greater than 2. Some of these polynomials can e factored completely using techniques learned in Chapter 4. KEY CONCEPT For Your Noteook Factoring Polynomials Definition A factorale polynomial with integer coefficients is factored completely if it is written as a product of unfactorale polynomials with integer coefficients. Examples 2(x 1 1)(x 2 4) and 5x 2 (x 2 2 ) are factored completely. x(x 2 2 4) is not factored completely ecause x 2 2 4 can e factored as (x 1 2)(x 2 2). E XAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 1 2x 2 2 15x 5 x(x 2 1 2x 2 15) Factor common monomial. 5 x(x 1 5)(x 2 ) Factor trinomial.. 2y 5 2 18y 5 2y (y 2 2 9) Factor common monomial. 5 2y (y 1 )(y 2 ) Difference of two squares c. 4z 4 2 16z 1 16z 2 5 4z 2 (z 2 2 4z 1 4) Factor common monomial. 5 4z 2 (z 2 2) 2 Perfect square trinomial 5.4 Factor and Solve Polynomial Equations 5
FACTORING PATTERNS In part () 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 cues. KEY CONCEPT For Your Noteook Special Factoring Patterns Sum of Two Cues Example a 1 5 (a 1 )(a 2 2 a 1 2 ) 8x 1 27 5 (2x) 1 5 (2x 1 )(4x 2 2 6x 1 9) Difference of Two Cues Example a 2 5 (a 2 )(a 2 1 a 1 2 ) 64x 2 1 5 (4x) 2 1 5 (4x 2 1)(16x 2 1 4x 1 1) E XAMPLE 2 Factor the sum or difference of two cues Factor the polynomial completely. a. x 1 64 5 x 1 4 Sum of two cues 5 (x 1 4)(x 2 2 4x 1 16). 16z 5 2 250z 2 5 2z 2 (8z 2 125) Factor common monomial. 5 2z 2 F (2z) 2 5 G Difference of two cues 5 2z 2 (2z 2 5)(4z 2 1 10z 1 25) GUIDED PRACTICE for Examples 1 and 2 Factor the polynomial completely. 1. x 2 7x 2 1 10x 2. y 5 2 75y. 16 5 1 686 2 4. w 2 27 FACTORING BY GROUPING For some polynomials, you can factor y grouping pairs of terms that have a common monomial factor. The pattern for factoring y grouping is shown elow. ra 1 r 1 sa 1 s 5 r(a 1 ) 1 s(a 1 ) 5 (r 1 s)(a 1 ) E XAMPLE Factor y grouping AVOID ERRORS An expression is not factored completely until all factors, such as x 2 2 16, cannot e factored further. Factor the polynomial x 2 x 2 2 16x 1 48 completely. x 2 x 2 2 16x 1 48 5 x 2 (x 2 ) 2 16(x 2 ) Factor y grouping. 5 (x 2 2 16)(x 2 ) Distriutive property 5 (x 1 4)(x 2 4)(x 2 ) Difference of two squares 54 Chapter 5 Polynomials and Polynomial Functions
QUADRATIC FORM An expression of the form au 2 1 u 1 c, where u is any expression in x, is said to e in quadratic form. The factoring techniques you studied in Chapter 4 can sometimes e used to factor such expressions. E XAMPLE 4 Factor polynomials in quadratic form IDENTIFY QUADRATIC FORM The expression 16x 4 2 81 is in quadratic form ecause it can e written as u 2 2 81 where u 5 4x 2. Factor completely: (a) 16x 4 2 81 and () 2p 8 1 10p 5 1 12p 2. a. 16x 4 2 81 5 (4x 2 ) 2 2 9 2 Write as difference of two squares. 5 (4x 2 1 9)(4x 2 2 9) Difference of two squares 5 (4x 2 1 9)(2x 1 )(2x 2 ) Difference of two squares. 2p 8 1 10p 5 1 12p 2 5 2p 2 (p 6 1 5p 1 6) Factor common monomial. 5 2p 2 (p 1 )(p 1 2) Factor trinomial in quadratic form. GUIDED PRACTICE for Examples and 4 Factor the polynomial completely. 5. x 1 7x 2 2 9x 2 6 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 factorale quadratic equations. You can extend this technique to solve some higher-degree polynomial equations. E XAMPLE 5 TAKS PRACTICE: Multiple Choice What are the real-numer solutions of the equation 4x 5 1 216x 5 60x? A 0, 2,, 6 B 2, 0, C 0, Ï } 6, D 2, 2Ï } 6, 0, Ï } 6, Solution 4x 5 1 216x 5 60x Write original equation. AVOID ERRORS Do not divide each side of an equation y a variale or a variale expression, such as 4x. Doing so will result in the loss of solutions. 4x 5 2 60x 1 216x 5 0 Write in standard form. 4x(x 4 2 15x 2 1 54) 5 0 Factor common monomial. 4x(x 2 2 9)(x 2 2 6) 5 0 Factor trinomial. 4 x(x 1 )(x 2 )(x 2 2 6) 5 0 Difference of two squares x 5 0, x 52, x 5, x 5 Ï } 6, or x 52Ï } 6 Zero product property c The correct answer is D. A B C D GUIDED PRACTICE for Example 5 Find the real-numer solutions of the equation. 8. 4x 5 2 40x 1 6x 5 0 9. 2x 5 1 24x 5 14x 10. 227x 1 15x 2 526x 4 5.4 Factor and Solve Polynomial Equations 55
E XAMPLE 6 Solve a polynomial equation CITY PARK You are designing a marle asin that will hold a fountain for a city park. The asin s sides and ottom should e 1 foot thick. Its outer length should e twice its outer width and outer height. What should the outer dimensions of the asin e if it is to hold 6 cuic feet of water? ANOTHER WAY For alternative methods to solving the prolem in Example 6, turn to page 60 for the Prolem Solving Workshop. Solution Volume (cuic feet) 5 Interior length (feet) p Interior width (feet) p Interior height (feet) 6 5 (2x 2 2) p (x 2 2) p (x 2 1) 6 5 (2x 2 2)(x 2 2)(x 2 1) Write equation. 05 2x 2 8x 2 1 10x 2 40 Write in standard form. 05 2x 2 (x 2 4) 1 10(x 2 4) Factor y grouping. 05 (2x 2 1 10)(x 2 4) Distriutive property c The only real solution is x 5 4. The asin 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 asin s dimensions e if it is to hold 128 cuic feet of water and have outer length 6x, width x, and height x? 5.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 7, 2, and 61 5 TAKS PRACTICE AND REASONING Exs. 9, 41, 6, 64, 66, and 67 1. VOCABULARY The expression 8x 6 1 10x 2 is in? form ecause it can e written as 2u 2 1 5u 2 where u 5 2x. 2. WRITING What condition must the factorization of a polynomial satisfy in order for the polynomial to e factored completely? EXAMPLE 1 on p. 5 for Exs. 9 MONOMIAL FACTORS Factor the polynomial completely.. 14x 2 2 21x 4. 0 2 54 2 5. c 1 9c 2 1 18c 6. z 2 6z 2 2 72z 7. y 5 2 48y 8. 54m 5 1 18m 4 1 9m 9. TAKS REASONING What is the complete factorization of 2x 7 2 2x? A 2x (x 1 2)(x 2 2)(x 2 1 4) B 2x (x 2 1 2)(x 2 2 2) C 2x (x 2 1 4) 2 D 2x (x 1 2) 2 (x 2 2) 2 56 Chapter 5 Polynomials and Polynomial Functions
EXAMPLE 2 on p. 54 for Exs. 10 17 EXAMPLE on p. 54 for Exs. 18 2 EXAMPLE 4 on p. 55 for Exs. 24 29 EXAMPLE 5 on p. 55 for Exs. 0 41 SUM OR DIFFERENCE OF CUBES Factor the polynomial completely. 10. x 1 8 11. y 2 64 12. 27m 1 1 1. 125n 1 216 14. 27a 2 1000 15. 8c 1 4 16. 192w 2 17. 25z 1 20 FACTORING BY GROUPING Factor the polynomial completely. 18. x 1 x 2 1 x 1 1 19. y 2 7y 2 1 4y 2 28 20. n 1 5n 2 2 9n 2 45 21. m 2 m 2 1 9m 2 22. 25s 2 100s 2 2 s 1 4 2. 4c 1 8c 2 2 9c 2 18 QUADRATIC FORM Factor the polynomial completely. 24. x 4 2 25 25. a 4 1 7a 2 1 6 26. s 4 2 s 2 2 24 27. 2z 5 2 2z 28. 6m 6 1 12m 4 1 m 2 29. 15x 5 2 72x 2 108x ERROR ANALYSIS Descrie and correct the error in finding all real-numer solutions. 0. 8x 2 27 5 0 (2x 1 )(4x 2 1 6x 1 9) 5 0 x 52 } 2 1. x 2 48x 5 0 x(x 2 2 16) 5 0 x 2 2 16 5 0 x 524 or x 5 4 SOLVING EQUATIONS Find the real-numer solutions of the equation. 2. y 2 5y 2 5 0. 18s 5 50s 4. g 1 g 2 2 g 2 5 0 5. m 1 6m 2 2 4m 2 24 5 0 6. 4w 4 1 40w 2 2 44 5 0 7. 4z 5 5 84z 8. 5 1 15 2 1 12 526 9. x 6 2 4x 4 2 9x 2 1 6 5 0 40. 48p 5 5 27p 41. TAKS REASONING What are the real-numer solutions of the equation x 4 2 27x 2 1 9x 5 x? A 21, 0, B 2, 0, C 2, 0, } 1, D 2, 2} 1, 0, CHOOSING A METHOD Factor the polynomial completely using any method. 42. 16x 2 44x 2 2 42x 4. n 4 2 4n 2 2 60 44. 24 4 2 500 45. 6a 2 15a 2 1 84a 2 5 46. 18c 4 1 57c 2 10c 2 47. 2d 4 2 1d 2 2 45 48. 2x 5 2 108x 2 49. 8y 6 2 8y 4 2 10y 2 50. z 5 2 z 4 2 16z 1 48 GEOMETRY Find the possile value(s) of x. 51. Area 5 48 52. Volume 5 40 5. Volume 5 125π x 2 4 2x 2 5 x 14 2x x x 1 2 x 2 1 CHOOSING A METHOD Factor the polynomial completely using any method. 54. x y 6 2 27 55. 7ac 2 1 c 2 2 7ad 2 2 d 2 56. x 2n 2 2x n 1 1 57. CHALLENGE Factor a 5 2 2 a 2 4 1 2a 4 2 2a 1 a 2 2 completely. 5.4 Factor and Solve Polynomial Equations 57
PROBLEM SOLVING EXAMPLE 6 on p. 56 for Exs. 58 6 58. ARCHAEOLOGY At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete lock with a volume of 945 cuic meters. The lock s dimensions are x meters high y 12x 2 15 meters long y 12x 2 21 meters wide. What is the height of the lock? LEBANON Caesarea SYRIA EGYPT ISRAEL 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 e in the ratio 1: : 6. What should the outer dimensions of the mold e if it is to hold 112 cuic centimeters of chocolate? 60. MULTI-STEP PROBLEM A production crew is assemling 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 cuic feet. 4x 6x 8x 2x 4x 6x x x x a. Write Expressions What is the volume, in terms of x, of each of the three levels of the platform?. Write an Equation Use what you know aout the total volume to write an equation involving x. c. Solve Solve the equation from part (). Use your solution to calculate the dimensions of each of the three levels of the platform. 61. SCULPTURE Suppose you have 250 cuic inches of clay with which to make a sculpture shaped as a rectangular prism. You want the height and width each to e 5 inches less than the length. What should the dimensions of the prism e? 62. MANUFACTURING A manufacturer wants to uild a rectangular stainless steel tank with a holding capacity of 670 gallons, or aout 89.58 cuic feet. The tank s walls will e one half inch thick, and aout 6.42 cuic feet of steel will e used for the tank. The manufacturer wants the outer dimensions of the tank to e related as follows: The width should e 2 feet less than the length. The height should e 8 feet more than the length. What should the outer dimensions of the tank e? x x 1 8 x 2 2 58 5 WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING
6. TAKS REASONING A platform shaped like a rectangular prism has dimensions x 2 2 feet y 2 2x feet y x 1 4 feet. Explain why the volume of the platform cannot e } 7 cuic feet. 64. TAKS REASONING In 2000 B.C., the Baylonians solved polynomial equations using tales of values. One such tale gave values of y 1 y 2. To e ale to use this tale, the Baylonians sometimes had to manipulate the equation, as shown elow. a x } ax 1 x 2 5 c 1 a2 x 2 } 2 5 a2 c } Original equation Multiply each side y a2 }. 1} ax 2 1 1} ax 2 2 5 a2 c } Rewrite cues and squares. They then found a2 c } in the y 1 y 2 column of the tale. Because the corresponding y-value was y 5 } ax, they could conclude that x 5 } y. a a. Calculate y 1 y 2 for y 5 1, 2,,..., 10. Record the values in a tale.. Use your tale and the method descried aove to solve x 1 2x 2 5 96. c. Use your tale and the method descried aove to solve x 1 2x 2 5 512. d. How can you modify the method descried aove for equations of the form ax 4 1 x 5 c? 65. CHALLENGE Use the diagram to complete parts (a) (c). a. Explain why a 2 is equal to the sum of the volumes of solid I, solid II, and solid III.. Write an algeraic expression for the volume of each of the three solids. Leave your expressions in factored form. c. Use the results from parts (a) and () to derive the factoring pattern for a 2 given on page 54. II I a III a a REVIEW Lesson 2.1; TAKS Workook REVIEW TAKS Preparation p. 408; TAKS Workook MIXED REVIEW FOR TAKS 66. TAKS PRACTICE Which inequality est descries the range of the function represented y the graph shown? TAKS Oj. 2 A y B y C 2 y D 24 y 4 67. TAKS PRACTICE A poster is shaped like an equilateral triangle with a side length of 0 inches. What is the approximate area of the poster? TAKS Oj. 8 F 195 in. 2 G 18 in. 2 H 90 in. 2 J 780 in. 2 TAKS PRACTICE at classzone.com 2 1 2 21 21 1 x 22 2 y 0 in. EXTRA PRACTICE for Lesson 5.4, p. 1014 ONLINE QUIZ at classzone.com 59
LESSON 5.4 TEKS a.5, a.6, 2A.2.A; P..B Using ALTERNATIVE METHODS Another Way to Solve Example 6, page 56 MULTIPLE REPRESENTATIONS In Example 6 on page 56, you solved a polynomial equation y factoring. You can also solve a polynomial equation using a tale or a graph. P ROBLEM CITY PARK You are designing a marle asin that will hold a fountain for a city park. The asin s sides and ottom should e 1 foot thick. Its outer length should e twice its outer width and outer height. What should the outer dimensions of the asin e if it is to hold 6 cuic feet of water? M ETHOD 1 Using a Tale One alternative approach is to write a function for the volume of the asin and make a tale of values for the function. Using the tale, you can find the value of x that makes the volume of the asin 6 cuic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the asin can hold is given y this function: y5 (2x 2 2)(x 2 2)(x 2 1) STEP 2 Make a tale of values for the function. Use only positive values of x ecause the asin s dimensions must e positive. STEP Identify the value of x for which y 5 6. The tale shows that y 5 6 when x 5 4. X 1 2 4 5 Y1=96 0 0 8 6 96 Y1 X 1 2 4 5 Y1=96 0 0 8 6 96 Y1 c The volume of the asin is 6 cuic feet when x is 4 feet. So, the outer dimensions of the asin should e as follows: Length 5 2x 5 8 feet Width 5 x 5 4 feet Height 5 x 5 4 feet 60 Chapter 5 Polynomials and Polynomial Functions
M ETHOD 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 asin 6 cuic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the asin can hold is given y this function: y5 (2x 2 2)(x 2 2)(x 2 1) STEP 2 Graph the equations y 5 6 and y 5 (x 2 1)(2x 2 2)(x 2 2). Choose a viewing window that shows the intersection of the graphs. STEP Identify the coordinates of the intersection point. On a graphing calculator, you can use the intersect feature. The intersection point is (4, 6). Intersection X=4 Y=6 c The volume of the asin is 6 cuic feet when x is 4 feet. So, the outer dimensions of the asin should e as follows: Length 5 2x 5 8 feet Width 5 x 5 4 feet Height 5 x 5 4 feet P RACTICE SOLVING EQUATIONS Solve the polynomial equation using a tale or using a graph. 1. x 1 4x 2 2 8x 5 96 2. x 2 9x 2 2 14x 1 7 52. 2x 2 11x 2 1 x 1 5 5 59 4. x 4 1 x 2 15x 2 2 8x 1 6 5245 5. 2x 4 1 2x 1 6x 2 1 17x 2 4 5 2 6. 2x 4 1 4x 1 8x 2 1 4x 2 11 5 1 7. 4x 4 2 16x 1 29x 2 2 95x 52150 8. WHAT IF? In the prolem on page 60, suppose the asin is to hold 200 cuic feet of water. Find the outer dimensions of the asin using a tale and using a graph. 9. PACKAGING A factory needs a ox that has a volume of 1728 cuic inches. The width should e 4 inches less than the height, and the length should e 6 inches greater than the height. Find the dimensions of the ox using a tale 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 e modeled y the function P(x) 5 0.0000984x 4 2 0.00712x 1 0.162x 2 2 1.11x 1 12. where x is the numer of years since 1970. In what year was the pineapple consumption aout 9.97 pounds per person? Solve the prolem using a tale and a graph. Using Alternative Methods 61