Find Rational Zeros. has integer coefficients, then every rational zero of f has the following form: x 1 a 0. } 5 factor of constant term a 0

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1 .6 Find Rational Zeros TEKS A.8.B; P..D, P..A, P..B Before You found the zeros of a polnomial function given one zero. Now You will find all real zeros of a polnomial function. Wh? So ou can model manufacturing processes, as in E.. Ke Vocabular zero of a function, p. constant term, p. 7 leading coefficient, p. 7 The polnomial function f() has }, }, and } 7 as 8 its zeros. Notice that the numerators of these zeros (,, and 7) are factors of the constant term, 0. Also notice that the denominators (,, and 8) are factors of the leading coefficient, 6. These observations are generalized b the rational zero theorem. KEY CONCEPT For Your Notebook The Rational Zero Theorem If f() a n n... a a 0 has integer coefficients, then ever rational zero of f has the following form: p } factor of constant term a 0 }}}}}}}}}}}}}} q factor of leading coefficient a n E XAMPLE List possible rational zeros List the possible rational zeros of f using the rational zero theorem. AVOID ERRORS Be sure our lists include both the positive and negative factors of the constant term and the leading coefficient. a. f() Factors of the constant term: 6, 6, 6, 6, 66, 6 Factors of the leading coefficient: 6 Possible rational zeros: 6 }, 6 }, 6 }, 6 }, 6 6 }, 6 }} Simplified list of possible zeros: 6, 6,6, 6, 66,6 b. f() 9 0 Factors of the constant term: 6, 6, 6, 60 Factors of the leading coefficient: 6, 6, 6 Possible rational zeros: 6}, 6}, 6}, 6}} 0, 6}, 6}, 6}, 6}} 0, 6}, 6}, 6}, 6}} 0 Simplified list of possible zeros: 6, 6, 6, 60, 6}, 6}, 6}, 6} 70 Chapter Polnomials and Polnomial Functions

2 GUIDED PRACTICE for Eample List the possible rational zeros of f using the rational zero theorem.. f() 9. f() 6 VERIFYING ZEROS In Lesson., ou found zeros of polnomial functions when one zero was known. The rational zero theorem is a starting point for finding zeros when no zeros are known. However, the rational zero theorem lists onl possible zeros. In order to find the actual zeros of a polnomial function f, ou must test values from the list of possible zeros. You can test a value b evaluating f() using the test value as. E XAMPLE Find zeros when the leading coefficient is Find all real zeros of f() 8 0. AVOID ERRORS Notice that not ever possible zero generated b the rational zero theorem is an actual zero of f. Solution STEP STEP List the possible rational zeros. The leading coefficient is and the constant term is 0. So, the possible rational zeros are: 6}, 6}, 6}, 6}, 6}} 0, 6}} 0 Test these zeros using snthetic division. Test : Test : is not a zero. is a zero. Because is a zero of f, ou can write f() ( )( 9 0). STEP Factor the trinomial in f() and use the factor theorem. f() ( )( 9 0) ( )( )( ) c The zeros of f are,, and. at classzone.com GUIDED PRACTICE for Eample Find all real zeros of the function.. f() 8. f() 8 LIMITING THE SEARCH FOR ZEROS In Eample, the leading coefficient of the polnomial function is. When the leading coefficient is not, the list of possible rational zeros can increase dramaticall. In such cases, the search can be shortened b sketching the function s graph..6 Find Rational Zeros 7

3 E XAMPLE Find zeros when the leading coefficient is not Find all real zeros of f() 0 7. Solution STEP List the possible rational zeros of f: 6}, 6}, 6}, 6}, 6} 6, 6}}, 6}, 6}, 6}, 6}, 6}, 6}, 6} 6, 6}}, 6}}, 6}} 0 0 STEP Choose reasonable values from the list above to check using the graph of the function. For f, the values }, }, }, and }} are reasonable based on the graph shown at the right. STEP Check the values using snthetic division until a zero is found. } } }} 69 }} }} } } is a zero. STEP Factor out a binomial using the result of the snthetic division. f() } (0 6 ) } ()( 8 7 ) Write as a product of factors. Factor out of the second factor. ( )( 8 7 ) Multipl the first factor b. STEP Repeat the steps above for g() 8 7. An zero of g will also be a zero of f. The possible rational zeros of g are: 6, 6, 6, 6, 66, 6, 6}, 6}, 6}, 6}, 6} 6, 6}} The graph of g shows that } ma be a zero. Snthetic division shows that } is a zero and g() } ( 0) ( )( ). It follows that: f() ( ) p g() ( )( )( ) STEP 6 Find the remaining zeros of f b solving 0. () 6 Ï}} () ()() }}}}}}}}}}}}} () 6 Ï} 7 }}}} Substitute for a, for b, and for c in the quadratic formula. Simplif. c The real zeros of f are }, }, Ï} 7 }}}}, and Ï} 7 }}}}. 7 Chapter Polnomials and Polnomial Functions

4 GUIDED PRACTICE for Eample Find all real zeros of the function.. f() f() 8 9 E XAMPLE TAKS Solve a REASONING: multi-step problem Multi-Step Problem ICE SCULPTURES Some ice sculptures are made b filling a mold with water and then freezing it. You are making such an ice sculpture for a school dance. It is to be shaped like a pramid with a height that is foot greater than the length of each side of its square base. The volume of the ice sculpture is cubic feet. What are the dimensions of the mold? Solution STEP Write an equation for the volume of the ice sculpture. Volume (cubic feet) } p Area of base (square feet) p Height (feet) } p p ( ) } ( ) Write equation. Multipl each side b and simplif. 0 Subtract from each side. STEP List the possible rational solutions: 6 }, 6 }, 6 }, 6 }, 6 6 }, 6 }} STEP STEP Test possible solutions. Onl positive -values make sense is a solution. Check for other solutions. The other two solutions, which satisf 6 0, are 6 iï} }}}}} and can be discarded because the are imaginar numbers. c The onl reasonable solution is. The base of the mold is feet b feet. The height of the mold is feet. GUIDED PRACTICE for Eample 7. WHAT IF? In Eample, suppose the base of the ice sculpture has sides that are foot longer than the height. The volume of the ice sculpture is 6 cubic feet. What are the dimensions of the mold?.6 Find Rational Zeros 7

5 .6 EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS on p. WS for Es. 7,, and 7 TAKS PRACTICE AND REASONING Es., 8, 9, 0, 0,, and. VOCABULARY Cop and complete: If a polnomial function has integer coefficients, then ever rational zero of the function has the form } p, q where p is a factor of the? and q is a factor of the?.. WRITING Describe a method ou can use to shorten the list of possible rational zeros when using the rational zero theorem. EXAMPLE on p. 70 for Es. 0 EXAMPLE on p. 7 for Es. 8 EXAMPLE on p. 7 for Es. 9 LISTING RATIONAL ZEROS List the possible rational zeros of the function using the rational zero theorem.. f() 8. g() 0. f() h() 8 7. g() 8. f() 9. h() h() 6 FINDING REAL ZEROS Find all real zeros of the function.. f(). f() 6. g() 0. h() h() f() f() g() 6 0 ELIMINATING POSSIBLE ZEROS Use the graph to shorten the list of possible rational zeros of the function. Then find all real zeros of the function. 9. f() f() 8. f() 6 6. f() Chapter Polnomials and Polnomial Functions

6 . MULTIPLE TAKS REASONING CHOICE According to the rational zero theorem, which is not a possible zero of the function f() 0 9? A 9 B } C } D FINDING REAL ZEROS Find all real zeros of the function.. f() 8 8. g() h() 7. f() 8. f() 9 9. g() 0. g() 9. h() 7. h() 6. f() f(). h() ERROR ANALYSIS Describe and correct the error in listing the possible rational zeros of the function. 6. f() 7 7. f() 6 Possible zeros:,, 7, Possible zeros: 6, 6, 6, 66, 6}, 6}, 6}, 6} 6 8. OPEN-ENDED TAKS REASONING MATH Write a polnomial function f that has a leading coefficient of and has possible rational zeros according to the rational zero theorem. 9. MULTIPLE TAKS REASONING CHOICE Which of the following is not a zero of the function f() ? A } B } 8 C } D } 0. SHORT TAKS REASONING RESPONSE Let a n be the leading coefficient of a polnomial function f and a 0 be the constant term. If a n has r factors and a 0 has s factors, what is the largest number of possible rational zeros of f that can be generated b the rational zero theorem? Eplain our reasoning. MATCHING Find all real zeros of the function. Then match each function with its graph.. f(). g(). h() A. B. C.. CHALLENGE Is it possible for a cubic function to have more than three real zeros? Is it possible for a cubic function to have no real zeros? Eplain..6 Find Rational Zeros 7

7 PROBLEM SOLVING EXAMPLE on p. 7 for Es. 8. MANUFACTURING At a factor, molten glass is poured into molds to make paperweights. Each mold is a rectangular prism with a height inches greater than the length of each side of its square base. Each mold holds 6 cubic inches of molten glass. What are the dimensions of the mold? 6. SWIMMING POOL You are designing a rectangular swimming pool that is to be set into the ground. The width of the pool is feet more than the depth, and the length is feet more than the depth. The pool holds 000 cubic feet of water. What are the dimensions of the pool? GEOMETRY In Eercises 7 and 8, write a polnomial equation to model the situation. Then list the possible rational solutions of the equation. 7. A rectangular prism has edges of lengths,, and and a volume of. 8. A pramid has a square base with sides of length, a height of, and a volume of. 9. MULTI-STEP PROBLEM From 99 to 00, the amount of athletic equipment E (in millions of dollars) sold domesticall can be modeled b E(t) 0t 0t 0t 8,0 where t is the number of ears since 99. Use the following steps to find the ear when about $0,00,000,000 of athletic equipment was sold. a. Write a polnomial equation that can be used to find the answer. b. List the possible whole-number solutions of the equation in part (a) that are less than 0. c. Use snthetic division to determine which of the possible solutions in part (b) is an actual solution. Then calculate the ear which corresponds to the solution. 0. EXTENDED TAKS REASONING RESPONSE Since 990, the number of U.S. travelers to foreign countries F (in thousands) can be modeled b F(t) t 6t 08t 9t,96 where t is the number of ears since 990. Use the following steps to find the ear when there were about 6,00,000 travelers. a. Write a polnomial equation that can be used to find the answer. b. List the possible whole-number solutions of the equation in part (a) that are less than or equal to 0. c. Use snthetic division to determine which of the possible solutions in part (b) is an actual solution. d. Graph the function F(t) and eplain wh there are no other reasonable solutions. Then calculate the ear which corresponds to the solution. WORKED-OUT SOLUTIONS 76 Chapter Polnomials on p. WS and Polnomial Functions TAKS PRACTICE AND REASONING

8 . CHALLENGE You are building a pair of ramps for a loading platform. The left ramp is twice as long as the right ramp. If 0 cubic feet of concrete are used to build the two ramps, what are the dimensions of each ramp? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson.; TAKS Workbook REVIEW Skills Review Handbook p. 99; TAKS Workbook. TAKS PRACTICE An electronics store has a 0%-off sale on all DVD plaers. Which statement best represents the functional relationship between the sale price of a DVD plaer and the original price? TAKS Obj. A B C D The original price is dependent on the sale price. The sale price is dependent on the original price. The sale price and the original price are independent of each other. The relationship cannot be determined.. TAKS PRACTICE The area of a rectangle is s 8 t 7 square units. The length of the rectangle is s t 9 units. What is the width of the rectangle? TAKS Obj. F s t 8 units H s t 6 units G 0s t 8 units J 8s t 6 units QUIZ for Lessons..6 Factor the polnomial completel. (p. ) Divide using polnomial long division or snthetic division. (p. 6) 7. ( 8 ) ( ) 8. ( 7 6) ( 7) Find all real zeros of the function. (p. 70) 9. f() f() 6. f(). f() 0. LANDSCAPING You are a landscape artist designing a square patio that is to be made from 8 cubic feet of concrete. The thickness of the patio is. feet less than each side length. What are the dimensions of the patio? (p. 70) EXTRA PRACTICE for Lesson.6, p. 0 ONLINE QUIZ.6 at Find classzone.com Rational Zeros 77

9 Spreadsheet ACTIVITY Use after Lesson.6.6 Use the Location Principle TEKS a., a., a.6 TEXAS classzone.com Kestrokes QUESTION How can ou use the Location Principle to identif zeros of a polnomial function? You can use the following result, called the Location Principle, to help ou find zeros of polnomial functions: If f is a polnomial function and a and b are two numbers such that f(a) < 0 and f(b) > 0, then f has at least one real zero between a and b. E XAMPLE Find zeros of a polnomial function Find all real zeros of f() STEP Enter values for Enter into cell A. Enter 0 into cell A. Tpe A into cell A. Select cells A through A7, and use the fill down command to fill in values of. STEP Enter values for f() Enter f() into cell B. Enter 6*A^*A^7*A6 into cell B. Select cells B through B7, and use the fill down command to fill in the values of f(). 6 7 A 0 B 6 7 A 0 B f() STEP Use Location Principle The spreadsheet in Step shows that f() < 0 and f() > 0. So, b the Location Principle, f has a zero between and. The rational zero theorem shows that the onl possible rational zero between and is }. Snthetic division confirms that } is a zero and that f can be factored as: f() } (6 ) ( )( 7 ) ( )( )( ) c The zeros of f are }, }, and. PRACTICE Find all real zeros of the function.. f() f() f() f() Chapter Polnomials and Polnomial Functions