Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.


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1 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. Type Example General trinomial 2x 2 2 3x (2x 1 5)(x 2 4) Perfect square trinomial x 2 1 8x (x 1 4) 2 Difference of two squares 9x (3x 1 1)(3x 2 1) Common monomial factor 8x x 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. KEY CONCEPT For Your Notebook Factoring Polynomials Definition A factorable polynomial with integer coefficients is factored completely if it is written as a product of unfactorable polynomials with integer coefficients. Examples 2(x 1 1)(x 2 4) and 5x 2 (x 2 2 3) are factored completely. 3x(x 2 2 4) is not factored completely because x can be factored as (x 1 2)(x 2 2). E XAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 3 1 2x x 5 x(x 2 1 2x 2 15) Factor common monomial. 5 x(x 1 5)(x 2 3) Factor trinomial. b. 2y y 3 5 2y 3 (y 2 2 9) Factor common monomial. 5 2y 3 (y 1 3)(y 2 3) Difference of two squares c. 4z z z 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 353
2 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 a 3 1 b 3 5 (a 1 b)(a 2 2 ab 1 b 2 ) 8x (2x) (2x 1 3)(4x 2 2 6x 1 9) Difference of Two Cubes Example a 3 2 b 3 5 (a 2 b)(a 2 1 ab 1 b 2 ) 64x (4x) (4x 2 1)(16x 2 1 4x 1 1) E XAMPLE 2 Factor the sum or difference of two cubes Factor the polynomial completely. a. x x Sum of two cubes 5 (x 1 4)(x 2 2 4x 1 16) b. 16z z 2 5 2z 2 (8z ) Factor common monomial. 5 2z 2 F (2z) G Difference of two cubes 5 2z 2 (2z 2 5)(4z z 1 25) GUIDED PRACTICE for Examples 1 and 2 Factor the polynomial completely. 1. x 3 2 7x x 2. 3y y b b 2 4. w FACTORING BY GROUPING For some polynomials, you can factor by grouping pairs of terms that have a common monomial factor. The pattern for factoring 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) E XAMPLE 3 Factor by grouping AVOID ERRORS An expression is not factored completely until all factors, such as x , cannot be factored further. Factor the polynomial x 3 2 3x x 1 48 completely. x 3 2 3x x x 2 (x 2 3) 2 16(x 2 3) Factor by grouping. 5 (x )(x 2 3) Distributive property 5 (x 1 4)(x 2 4)(x 2 3) Difference of two squares 354 Chapter 5 Polynomials and Polynomial Functions
3 QUADRATIC FORM An expression of the form au 2 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. E XAMPLE 4 Factor polynomials in quadratic form IDENTIFY QUADRATIC FORM The expression 16x is in quadratic form because it can be written as u where u 5 4x 2. Factor completely: (a) 16x and (b) 2p p p 2. a. 16x (4x 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 3)(2x 2 3) Difference of two squares b. 2p p p 2 5 2p 2 (p 6 1 5p 3 1 6) Factor common monomial. 5 2p 2 (p 3 1 3)(p 3 1 2) Factor trinomial in quadratic form. GUIDED PRACTICE for Examples 3 and 4 Factor the polynomial completely. 5. x 3 1 7x 2 2 9x g t t t 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 higherdegree polynomial equations. E XAMPLE 5 Standardized Test Practice What are the realnumber solutions of the equation 3x x 5 18x 3? A 0, 1, 3, 5 B 21, 0, 1 C 0, 1, Ï 5 D 2 Ï 5, 21, 0, 1, Ï 5 Solution 3x x 5 18x 3 Write original equation. 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. 3x x x 5 0 Write in standard form. 3x(x 4 2 6x 2 1 5) 5 0 Factor common monomial. 3x(x 2 2 1)(x 2 2 5) 5 0 Factor trinomial. 3 x(x 1 1)(x 2 1)(x 2 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 GUIDED PRACTICE for Example 5 Find the realnumber solutions of the equation. 8. 4x x x x x 5 14x x x x Factor and Solve Polynomial Equations 355
4 E XAMPLE 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? ANOTHER WAY For alternative methods to solving the problem in Example 6, turn to page 360 for the Problem Solving Workshop. 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 x 3 2 8x x 2 40 Write in standard form x 2 (x 2 4) 1 10(x 2 4) Factor by grouping. 0 5 (2x )(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 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 SKILL PRACTICE HOMEWORK KEY 5 WORKEDOUT SOLUTIONS on p. WS1 for Exs. 7, 23, and 61 5 STANDARDIZED TEST PRACTICE Exs. 2, 9, 41, 63, and VOCABULARY The expression 8x x is in? form because it can be written as 2u 2 1 5u 2 3 where u 5 2x 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 x x 4. 30b b 2 5. c 3 1 9c c 6. z 3 2 6z z 7. 3y y m m 4 1 9m 3 9. MULTIPLE CHOICE What is the complete factorization of 2x x 3? A 2x 3 (x 1 2)(x 2 2)(x 2 1 4) B 2x 3 (x 2 1 2)(x 2 2 2) C 2x 3 (x 2 1 4) 2 D 2x 3 (x 1 2) 2 (x 2 2) Chapter 5 Polynomials and Polynomial Functions
5 EXAMPLE 2 on p. 354 for Exs EXAMPLE 3 on p. 354 for Exs EXAMPLE 4 on p. 355 for Exs EXAMPLE 5 on p. 355 for Exs SUM OR DIFFERENCE OF CUBES Factor the polynomial completely. 10. x y m n a c w z FACTORING BY GROUPING Factor the polynomial completely. 18. x 3 1 x 2 1 x y 3 2 7y 2 1 4y n 3 1 5n 2 2 9n m 3 2 m 2 1 9m s s 2 2 s c 3 1 8c 2 2 9c 2 18 QUADRATIC FORM Factor the polynomial completely. 24. x a 4 1 7a s 4 2 s z 5 2 2z m m 4 1 m x x x ERROR ANALYSIS Describe and correct the error in finding all realnumber solutions x (2x 1 3)(4x 2 1 6x 1 9) 5 0 x x x 5 0 3x(x ) 5 0 x x 5 24 or x 5 4 SOLVING EQUATIONS Find the realnumber solutions of the equation. 32. y 3 2 5y s s 34. g 3 1 3g 2 2 g m 3 1 6m 2 2 4m w w z z b b b x 6 2 4x 4 2 9x p p MULTIPLE CHOICE What are the realnumber solutions of the equation 3x x 2 1 9x 5 x 3? 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 x x x 43. n 4 2 4n b b a a a c c c d d x x y y y z 5 2 3z z 1 48 GEOMETRY Find the possible value(s) of x. 51. Area Volume Volume 5 125π x 2 4 2x 2 5 x 1 4 2x 3x 3x 1 2 x 2 1 CHOOSING A METHOD Factor the polynomial completely using any method. 54. x 3 y ac 2 1 bc 2 2 7ad 2 2 bd x 2n 2 2x n CHALLENGE Factor a 5 b 2 2 a 2 b 4 1 2a 4 b 2 2ab 3 1 a 3 2 b 2 completely. 5.4 Factor and Solve Polynomial Equations 357
6 PROBLEM SOLVING EXAMPLE 6 on p. 356 for Exs 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 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 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. MULTISTEP PROBLEM A production crew is assembling a threelevel 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. 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? 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 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? x x 1 8 x WORKEDOUT SOLUTIONS 358 Chapter 5 Polynomials on p. WS1 and Polynomial Functions 5 STANDARDIZED TEST PRACTICE
7 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 cubic feet 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. ax 3 1 bx 2 5 c a 3 x 3 1 a2 x 2 b 3 b 2 1 ax b ax 5 a2 c b 3 b a2 c b 3 Original equation Multiply each side by a2 b 3. Rewrite cubes and squares. They then found a2 c b in the 3 y3 1 y 2 column of the table. Because the corresponding yvalue was y 5 ax by, they could conclude that x 5 b a. 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 2x c. Use your table and the method described above to solve 3x 3 1 2x d. How can you modify the method described above for equations of the form ax 4 1 bx 3 5 c? 65. CHALLENGE Use the diagram to complete parts (a) (c). a. Explain why a 3 2 b 3 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 a 3 2 b 3 given on page 354. II b I a b III b a a MIXED REVIEW Graph the function. 66. f(x) 5 22 x (p. 123) 67. y x2 1 4x 1 5 (p. 236) 68. y 5 3(x 1 4) (p. 245) 69. f(x) 5 x 3 2 2x 2 5 (p. 337) Graph the inequality in a coordinate plane. (p. 132) 70. y 2x y > 25 2 x 72. y < 0.5x x 1 12y x 2 9y x y > 5 PREVIEW Prepare for Lesson 5.5 in Exs Use synthetic substitution to evaluate the polynomial function for the given value of x. (p. 337) 76. f(x) 5 5x 4 2 3x 3 1 4x 2 2 x 1 10; x f(x) 5 23x 5 1 x 3 2 6x 2 1 2x 1 4; x f(x) 5 5x 5 2 4x x ; x f(x) 5 26x 4 1 9x 2 15; x 5 4 EXTRA PRACTICE for Lesson 5.4, p ONLINE Factor and QUIZ Solve Polynomial at classzone.com Equations 359
8 LESSON 5.4 Using ALTERNATIVE METHODS 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. P ROBLEM 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? M ETHOD 1 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. STEP 3 Identify the value of x for which y The table shows that y 5 36 when x 5 4. X Y1= Y1 X Y1= Y1 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 Chapter 5 Polynomials and Polynomial Functions
9 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 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 window that 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). 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 RACTICE SOLVING EQUATIONS Solve the polynomial equation using a table or using a graph. 1. x 3 1 4x 2 2 8x x 3 2 9x x x x 2 1 3x x 4 1 x x 2 2 8x x 4 1 2x 3 1 6x x x 4 1 4x 3 1 8x 2 1 4x x x x x 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) x x x x where x is the number of years since 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 361
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