CHAPTER 6 Vocabular The table contains important vocabular terms from Chapter 6. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. Term Page Definition Clarifing Eample degree of a monomial degree of a polnomial end behavior leading coefficient local maimum local minimum 120 Algebra 2
CHAPTER 6 Vocabular The table contains important vocabular terms from Chapter 6. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. degree of a monomial degree of a polnomial end behavior leading coefficient Term Page Definition Clarifing Eample local maimum 06 06 53 06 55 The sum of the eponents of the variables in the monomial. The degree of the term of the polnomial with the greatest degree. The trends in the -values of a function as the -values approach positive and negative infinit. The coefficient of the first term of a polnomial in standard form. For a function f, f(a) is a local maimum if there is an interval around a such that f() f(a) for ever -value in the interval ecept a. 2 5 z 3 Degree: 2 5 3 10 5 5 0 Degree: 0 3 2 2 5 12 3 2 Degree Degree 6 Degree 5 6 is the degree of this polnomial. End behavior: f() as f() as 3 2 7 2 Leading coefficient: 3 0 0 120 160 local maimum local minimum 55 For a function f, f(a) is a local minimum if there is an interval around a such that f() f(a) for ever -value in the interval ecept a. 0 0 120 160 local minimum 120 Algebra 2
CHAPTER 6 VOCABULARY CONTINUED monomial Term Page Definition Clarifing Eample multiplicit polnomial snthetic division turning point 121 Algebra 2
CHAPTER 6 VOCABULARY CONTINUED Term Page Definition Clarifing Eample monomial multiplicit polnomial snthetic division turning point 06 39 06 23 55 A number or a product of numbers and variables with wholenumber eponents, or a polnomial with one term. If a polnomial P() has a multiple root at r, the multiplicit of r is the number of times ( r) appears as a factor in P(). A monomial or a sum or difference of monomials. A shorthand method of dividing b a linear binomial of the form ( a) b writing onl the coefficients of the polnomials. A point on the graph of a function that corresponds to a local maimum (or minimum) where the graph changes from increasing to decreasing (or vice versa). 3 2 P() ( 2)3( 3) has a root of multiplicit 3 at 2. 2 2 3 7 2 Divide (2 2 7 9) b ( 2) 2 2 7 9 2 2 2 7 9 2 3 2 2 7 9 6 2 3 3 3 2 3 2 20 10 0 turning point 121 Algebra 2
CHAPTER 6 Chapter Review 6-1 Polnomials Rewrite each polnomial in standard form. Then identif the leading coefficient, degree, and number of terms. Name the polnomial. 1. 3 2 3 3 2 2. 1 3 2 Leading coefficient: Degree: Number of Terms: Name: Leading coefficient: Degree: Number of Terms: Name: 3. 2 3 3 3 5. 22 6 Leading coefficient: Degree: Number of Terms: Name: Leading coefficient: Degree: Number of Terms: Name: Add or subtract. Write our answer in standard form. 5. (2 2 3 6) ( 5 2 6) 6. (3 3 2 ) (6 3 3 2 7) 7. (1 2 2 ) ( 7 5 9 2 ). The cost on -units of a product can be modeled b C() 3 1 12. Evaluate C() for 50, and describe what the value represents. Graph each polnomial function on a calculator. Describe the graph, and identif the number of real zeros. 9. g() 3 7 6 137 Algebra 2
CHAPTER 6 Chapter Review 6-1 Polnomials Rewrite each polnomial in standard form. Then identif the leading coefficient, degree, and number of terms. Name the polnomial. 1. 3 2 3 3 2 2. 1 3 2 Leading coefficient: 3 Degree: Number of Terms: Name: quartic polnomial with four terms Leading coefficient: 3 Degree: 2 Number of Terms: 3 Name: quadratic trinomial 3. 2 3 3 3 5. 22 6 Leading coefficient: 3 Degree: 5 Number of Terms: Name: quintic polnomial with four terms Leading coefficient: 6 Degree: 1 Number of Terms: 2 Name: linear binomial Add or subtract. Write our answer in standard form. 5. (2 2 3 6) ( 5 2 6) 6. (3 3 2 ) (6 3 3 2 7) 3 2 7. (1 2 2 ) ( 7 5 9 2 ) 3 3 11 2 7 10 2 3 7. The cost on -units of a product can be modeled b C() 3 1 12. Evaluate C() for 50, and describe what the value represents. Graph each polnomial function on a calculator. Describe the graph, and identif the number of real zeros. 9. g() 3 7 6 C(50) 12,112; The cost of manufacturing 50 units is $12,112. From left to right, the graph increases then decreases, then increases. It crosses the -ais 3 times at 2, 1 and 3. 137 Algebra 2
Graph each polnomial function on a calculator. Describe the graph, and identif the number of real zeros. 10. h() 5 11. f() 3 2 6-2 Multipling Polnomials Find each product. 12. 7( 3 ) 13. ( )( 2 2 ) 1. 1 2 5 2 15. (3 2)(5 2 6) Epand each epression. 16. ( 5) 3 17. ( 2) 1. (2 1) 19. Find the polnomial epression in terms of for the volume of the rectangular prism shown. + 2 2 3 1 13 Algebra 2
Graph each polnomial function on a calculator. Describe the graph, and identif the number of real zeros. 10. h() 5 From left to right, the graph decreases. It crosses the -ais 3 times at 1, 0 and 1.decreases. It crosses the -ais 3 times at 1, 0 and 1.decreases, then increases, then decreases. It crosses the -ais 3 times at 1, 0 and 1. 11. f() 3 2 From left to right the graph decreases, then increases. It crosses the -ais twice at 1 and 2. 6-2 Multipling Polnomials Find each product. 12. 7( 3 ) 13. ( )( 2 2 ) 56 2 2 3 2 2 3 1. 1 2 5 2 15. (3 2)(5 2 6) 1 2 5 25 15 3 3 2 10 2 1 2 12 Epand each epression. 16. ( 5) 3 17. ( 2) 3 15 2 75 125 3 2 2 2 32 3 16 1. (2 1) 16 32 3 2 2 1 19. Find the polnomial epression in terms of for the volume of the rectangular prism shown. 2 3 2 + 2 2 3 1 13 Algebra 2
6-3 Dividing Polnomials Divide. 20. (1 2 3 10) (3 2) 21. (2 3 1 2 33 35) ( 7) Use snthetic substitution to evaluate the polnomial for the given value. 22. P() 3 9 2 3 7 23. P() 3 10 2 10 5 for 2 for 1 6- Factoring Polnomials Factor each epression. 2. 9 2 25 25. 2 3 2 2 26. a 3 6a 2 3a 1 27. t 9 6 2. The volume of a bo is modeled b the function V() 3 2 7 10. Identif the values of for which the volume is 0 and use the graph to factor V(). 12 2 12 16 0 6-5 Finding Real Roots of Polnomial Equations 29. The earl profit of a compan in thousands of dollars can be modeled b P(t) 3 2 225, where t is the number of ears since 1999. Factor to find the ears in which the profit was 0. 139 Algebra 2
6-3 Dividing Polnomials Divide. 20. (1 2 3 10) (3 2) 21. (2 3 1 2 33 35) ( 7) 6 5 2 2 5 Use snthetic substitution to evaluate the polnomial for the given value. 22. P() 3 9 2 3 7 23. P() 3 10 2 10 5 for 2 for 1 15 27 6- Factoring Polnomials Factor each epression. 2. 9 2 25 25. 2 3 2 2 (3 5)(3 5) 2( 2)( 6) 26. a 3 6a 2 3a 1 27. t 9 6 (a 6)(a 2 3) (t 3 )(t 6 t 3 16) 2. The volume of a bo is modeled b the function V() 3 2 7 10. Identif the values of for which the volume is 0 and use the graph to factor V(). 12 2 2, 1, 5; V() ( 2)( 1)( 5) 12 16 0 6-5 Finding Real Roots of Polnomial Equations 29. The earl profit of a compan in thousands of dollars can be modeled b P(t) 3 2 225, where t is the number of ears since 1999. Factor to find the ears in which the profit was 0. 2002, 200 139 Algebra 2
Identif the roots of each equation. State the multiplicit of each root. 30. 3 3 2 72 31. 2 3 2 2 2 32. 2 3 7 2 32 0 0 0 6-6 Fundamental Theorem of Algebra Write the simplest polnomial function with the given roots. 33. 1, 2, 3 3. i, i, 0 6-7 Investigating Graphs of Polnomials Functions 35. Solve 5 3 2 20 16 0 b finding all roots. Graph each function. 36. f () 3 2 2 3 37. f() 3 3 20 10 6 2 6 2 15 10 5 5 10 15 0 5 2 10 Algebra 2
Identif the roots of each equation. State the multiplicit of each root. 30. 3 3 2 72 31. 2 3 2 2 2 32. 2 3 7 2 32 0 0 0 6 multiplicit of 2; 9 multiplicit of 1 2, 3, multiplicit of 1 0 multiplicit of 1; multiplicit of 1, 1 multiplicit of 2 6-6 Fundamental Theorem of Algebra Write the simplest polnomial function with the given roots. 33. 1, 2, 3 3. i, i, 0 3 2 6 3 6-7 Investigating Graphs of Polnomials Functions 35. Solve 5 3 2 20 16 0 b finding all roots. 1,, 2i Graph each function. 36. f () 3 2 2 3 37. f() 3 3 20 10 6 2 6 2 15 10 5 5 10 15 0 5 2 10 Algebra 2
Identif whether the function graphed has an odd or even degree and a positive or negative leading coefficient. 3. 39. 0. 6- Transforming Polnomial Functions Let f() 3 2 2 3 1. Write a function g() that performs each transformation. 1. Reflect f() across -ais. 2. Reflect f() across the -ais. Let f() 2 2 2 3. Graph f() and g() on the same coordinate plane. Describe g() as a transformation of f(). 7. g() 2f(). g() f( 2) 9. g() f(2) 2 12 12 2 2 11 Algebra 2
Identif whether the function graphed has an odd or even degree and a positive or negative leading coefficient. 3. 39. 0. odd; positive even; positive odd; negative 6- Transforming Polnomial Functions Let f() 3 2 2 3 1. Write a function g() that performs each transformation. 1. Reflect f() across -ais. 2. Reflect f() across the -ais. g() 3 2 2 3 1 g() 3 2 3 1 Let f() 2 2 2 3. Graph f() and g() on the same coordinate plane. Describe g() as a transformation of f(). 7. g() 2f(). g() f( 2) 9. g() f(2) 2 12 12 2 2 reflect across -ais verticall stretched b a factor of 2 shifted horizontall 2 units to the left horizontall compressed b a factor of 2 11 Algebra 2
6-9 Curve Fitting with Polnomial Models 6. The table shows the population of butterflies in a butterfl house. Write a polnomial function for the data. Time (h) 1 2 3 5 Number of butterflies 26 72 10 370 665 12 Algebra 2
6-9 Curve Fitting with Polnomial Models 6. The table shows the population of butterflies in a butterfl house. Write a polnomial function for the data. Time (h) 1 2 3 5 Number of butterflies 26 72 10 370 665 Cubic: f() 3.5 3 9.1 2 6. 19.6 or quartic: f() 0.125 2.03 3 15.37 2 16.53 25 12 Algebra 2
CHAPTER 6 Big Ideas Answer these questions to summarize the important concepts from Chapter 6 in our own words. 1. Eplain the importance of factoring polnomials. 2. Eplain how to tell if a function is increasing or decreasing. 3. Eplain the difference between the graphs of f() and f( a).. Eplain what is meant b epanding a polnomial ( a) b. For more review of Chapter 6: Complete the Chapter 6 Stud Guide and Review on pages 777 of our tetbook. Complete the Read to Go On quizzes on pages 37 and 73 of our tetbook. 13 Algebra 2
CHAPTER 6 Big Ideas Answer these questions to summarize the important concepts from Chapter 6 in our own words. 1. Eplain the importance of factoring polnomials. Answers will var. Possible answer: It enables ou to find roots or solutions of equations. 2. Eplain how to tell if a function is increasing or decreasing. Answers will var. Possible answer: If the leading coefficient is positive then the function is increasing. If the leading coefficient is negative, then the function is decreasing. 3. Eplain the difference between the graphs of f() and f( a). Answers will var. Possible answer: The graph of f( a) is translated a units to the left of f () on the coordinate sstem.. Eplain what is meant b epanding a polnomial ( a) b. Answers will var. Possible answer: To epand a polnomial means to multipl the factor a b itself b times and then simplif the product. For more review of Chapter 6: Complete the Chapter 6 Stud Guide and Review on pages 777 of our tetbook. Complete the Read to Go On quizzes on pages 37 and 73 of our tetbook. 13 Algebra 2