015 College Board. All rights reserved. Postal Service Write your answers on notebook paper. Show your work. Lesson 1-1 1. The volume of a rectangular bo is given by the epression V = (10 6w)w, where w is measured in inches. a. What is a reasonable domain for the function in this situation? Epress the domain as an inequality, in interval notation, and in set notation. b. Sketch a graph of the function over the domain that you found. Include the scale on each ais. c. Use a graphing calculator to find the coordinates of the maimum point of the function. d. What is the width of the bo, in inches, that produces the maimum volume?. A cylindrical can is being designed for a new product. The height of the can plus twice its radius must be 5 cm. a. Find an equation that represents the volume of the can, given the radius. b. Find the radius that yields the maimum volume. c. Find the maimum volume of the can. Lesson 1-3. Sketch the graph of the polynomial function f() = 3 +.. Name any - or y-intercepts of the function f() in Item 3. 5. Name any relative maimum values and relative minimum values of the function f() in Item 3. ACTIVITY 1 For Items 6 10, decide if each function is a polynomial. If it is, write the function in standard form, and then state the degree and leading coefficient. 6. f() = 7 9 3 + 3 7 7. f() = 3 + 5 + 9. f() = + + 5 1 3 9. f() = 0.3 3 + 0.0 + 5 1 3 10. f() = 3+ 5+ 11. Eamine the graph below. y Which of the following statements is NOT true regarding the polynomial whose graph is shown? A. The degree of the polynomial is even. B. The leading coefficient is positive. C. The function is a second-degree polynomial. D. As ±, y. ACTIVITY 1 Continued ACTIVITY PRACTICE 1. a. 0 < w < 0; (0, 0); {w w R, 0 < w < 0} b. V(w) 000 7000 6000 5000 000 3000 000 1000 1 16 0 w c. (13.333, 7111.111) d. about 13.3 inches. a. V = πr 3 + 5πr b. 15 cm c. about 10,603 cm 3 3. y 1 1. -intercepts: 0 and ; y-intercept: 0 5. relative maimum value: 0; relative minimum value: 1.15 6. Yes; f() = 3 7 9 3 + 7 ; seventh degree; 3 7. No.. Yes; f( )= 1 3 + + 5; fourth degree; 1 9. No. 10. No. 11. C Activity 1 Introduction to Polynomials 39
ACTIVITY 1 Continued 1. As ±, f( ). 13. As, f( ), and as, f( ). 1. Polynomials are continuous functions. Since one side of the graph increases without bound and the other side decreases without bound, the graph must cross the -ais in at least one place. 15. Check students work. 16. even 17. neither 1. odd 19. B 0. Check students work. 1. ( 5, 3); Since an even function is symmetric over the y-ais, you can reflect the point (5, 3) over the y-ais to get the point ( 5, 3).. Sharon is correct that the function is a polynomial function and that it has a positive leading coefficient. However, the function is not an even function because it is not symmetric over the y-ais. She is also incorrect about the degree; since the graph crosses the -ais four times, it must be at least a fourth-degree polynomial. ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems. ACTIVITY 1 For Items 1 and 13, describe the end behavior of each function using arrow notation. 1. f() = 6 3 + 3 + 13. f() = 3 + 7 11 1. Use the concept of end behavior to eplain why a third-degree polynomial function must have at least one -intercept. 15. Sketch a graph of any third-degree polynomial function that has eactly one -intercept, a relative minimum at (, 1), and a relative maimum at (, 3). Lesson 1-3 For Items 16, determine whether each function is even, odd, or neither. 16. f() = 10 + 3 17. f() = 3 + + 5 1. f() = 6 5 19. When graphed, which of the following polynomial functions is symmetric about the origin? A. f() = 3 + + 5 B. f() = 3 + C. f() = 7 + 5 D. f() = 5 3 + 3 7 + 1 Postal Service 0. Sketch a graph of an even function whose degree is greater than. 1. If f() is an even function and passes through the point (5, 3), what other point must lie on the graph of the function? Eplain your reasoning. MATHEMATICAL PRACTICES Construct Viable Arguments and Critique the Reasoning of Others. Sharon described the function graphed below as follows: It is a polynomial function. It is an even function. It has a positive leading coefficient. The degree n could be any even number greater than or equal to. Critique Sharon s description. If you disagree with any of her statements, provide specific reasons as to why. y 015 College Board. All rights reserved. 0 SpringBoard Mathematics Algebra, Unit 3 Polynomials
Polynomial Operations Polly s Pasta ACTIVITY 15 ACTIVITY 15 Continued ACTIVITY PRACTICE 1. ACTIVITY 15 PRACTICE Write your answers on notebook paper. Show your work. Lesson 15-1 1. The graph below shows the number of visitors at a public library one day between the hours of 9:00 a.m. and 7:00 p.m. The round dots represent A(t), the number of adult visitors, and the diamonds represent C(t), the number of children and teenage visitors. Graph V(t), the total number of visitors, and eplain how you used the graph to find the values of V(t). Visitors 0 70 60 50 0 30 0 10 9:00 11:00 1:00 3:00 5:00 7:00 t Time (9:00 a.m. 7:00 p.m.). Eamine the functions graphed in Item 1. Which of the statements is true over the given domain of the functions? A. A(t) > C(t) B. C(t) > A(t) C. A(t) C(t) > 0 D. V(t) > C(t) 3. The polynomial epressions 5 + 7, 3 + 9, and 3 represent the lengths of the sides of a triangle for all whole-number values of > 1. Write an epression for the perimeter of the triangle.. In Item 3, what kind of epression is the perimeter epression? Lesson 15-5. An open bo will be made by cutting four squares of equal size from the corners of a 10-inch-by-1-inch rectangular piece of cardboard and then folding up the sides. The epression V() = (10 )(1 ) can be used to represent the volume of the bo. Write this epression as a polynomial in standard form. 6. Write an epression for the volume of a bo that is constructed in the same way as in Item 5, but from a rectangular piece of cardboard that measures inches by 1 inches. Write your epression in factored form, and then as a polynomial in standard form. 7. Write an epression to represent the combined volume of the two boes described in Items 5 and 6. For Items 13, find each sum or difference.. (3 ) + (5 + 1) 9. ( 6 + 5) ( + + 1) 10. ( 1 + 9) + (3 11) 11. (6 13 + ) ( 7 + 5) 1. ( 3 + 1) + (5 + ) 13. ( + 1) ( + 5 + 9) 0 70 60 50 0 30 0 10 9:0011:001:00 3:00 5:00 7:00 t Each value of V(t) is the sum of the values for A(t) and C(t) at the given t.. D 3. P( )= 6 + 3+ 16. a polynomial 3 5. V( )= + 10 6. V( ) = ( )( 1 ) ; 3 V( )= + 11 3 7. V( )= + 3. 3 9. 7 + 10. 9 11. 6 1 1. 3 + 5 + + 1 13. 6 015 College Board. All rights reserved. Activity 15 Polynomial Operations 53
ACTIVITY 15 Continued 1. 0 + 15 3 5 15. 0 + 5 16. 6 + 3 y 3 + y 6 17. 3 3 1 3 1 1. 15 3 + 9 + 1 19. A 0. 7 + 11 + 1 1. 5 + 16+ + 3+ 1. + + 6+ 5 3. + 0 +. 3 + + 0+ 5 5. 10+ 15 1 + 3 6. a. Yes; check students work. Students eamples should show the highest-order terms summing to zero. b. No; because you are combining like terms, there is no way for the degree to increase. ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems. ACTIVITY 15 For Items 1 1, find each product. Write your answer as a polynomial in standard form. 1. 5 ( + 3 9) 15. ( 5) 16. ( 3 + y 3 ) 17. ( + )(3 3 + 7) 1. ( 3)( 3 9 + 6) Lesson 15-3 19. Which of the following quotients CANNOT be found using synthetic division? A. 3 + + 5 + 1 B. + 1 1 5 C. + 10 + 50 3 D. + 1 Polly s Pasta For Items 0, find each quotient using long division. 0. 6 + + 1 1. (5 + 1 3 + 9) ( + 3 + 1). ( 3 3 + 7) ( ) For Items 3 5, find each quotient using synthetic division. 3. ( + ) ( + ) 3. 3 10 + 1 5. ( 3 15 + ) ( + 3) MATHEMATICAL PRACTICES Reason Abstractly and Quantitatively 6. Before answering parts a and b, review them carefully to ensure you understand all the terminology and what is being asked. a. When adding two polynomials, is it possible for the degree of the sum to be less than the degree of either of the polynomials being added (the addends)? If so, give an eample to support your answer. If not, eplain your reasoning. b. Is it possible for the degree of the sum to be greater than the degree of either of the addends? If so, give an eample to support your answer. If not, eplain your reasoning. 015 College Board. All rights reserved. 5 SpringBoard Mathematics Algebra, Unit 3 Polynomials
Pascal s Triangle Write your answers on notebook paper. Show your work. Lesson 16-1 1. Which of the following would you use to find the number of different combinations of si-person nominating committees that could be chosen from a class of 5 students? A. 6 C 6 5 =! 5!( 5 6!) B. 5 C 5 6 =! 5!( 5 6!) C. 5 C 5 6 =! 6!( 5 6!) D. 6 C 6 5 =! 6!( 5 6!). Simplify: 9 7 6 5 3 1 ( 6 5 3 1)( 3 1) 3. Write the epression in Item in n C r notation.. Find the number of different combinations of four-person nominating committees that could be chosen from a class of 5 students. 5. Write the numbers that will fill in the eighth row of Pascal s triangle. 6. In which row of Pascal s triangle would you find the coefficients for the terms in the epansion of (a + b) 1? 7. Which of the following has the same value as 1 7? A. 1 C 7 B. 1 C 5 C. 1 5 D. all of the above ACTIVITY 16. Use what you have learned about the patterns in Pascal s triangle to epand (a + b). 9. Manuela started epanding ( + y) 9. So far, she has written: 9 + 9 y + 36 7 y + 6 y 3 + 16 5 y + 16 y 5 Manuela eplained to Karen that since both coefficients in the binomial are 1, the coefficients of the terms will start repeating, only backwards. Use Manuela s strategy to complete the epansion. ACTIVITY 16 Continued ACTIVITY PRACTICE 1. C. 3. 9 C 6. 1,650 combinations 5. 1 7 1 35 1 7 1 6. 15th row 7. D. a + a 7 b + a 6 b + 56a 5 b 3 + 70a b + 56a 3 b 5 + a b 6 + ab 7 + b 9. 3 y 6 + 36 y 7 + 9y + y 9 015 College Board. All rights reserved. Activity 16 Binomial Theorem 63
ACTIVITY 16 Continued 10. 9 k= 0 9 9 k a b k k 7 7 7 k 11. ( ) ( 3 ) k k= 0 1. 60 13. D 1. 5 15. 30 3 16. + 0 3 + 150 + 500 + 65 17. 10a 5 + 10a b + 60a 3 b + 160a b 3 + 0ab + b 5 1. 5 15 + 90 3 70 + 05 3 19. 3 + 1 y + 6y + y 3 0. a. 10 + 5 + 10 6 + 10 + 5 + 1; This epansion disproves the statement since the powers of decrease by, not 1. b. Sample answer: In the epansion of every linear binomial, ( + a) n, the powers of decrease by 1 from left to right when written as a polynomial in standard form. ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems. k ACTIVITY 16 Lesson 16-10. Write (a + b) 9 using summation notation. 11. Write ( 3) 7 using summation notation. 1. Find the coefficient of the fourth term in the epansion of ( + ) 5. 13. Which of the following is the coefficient of the third term in the epansion of ( ) 7? A. B. 1 C. 1 D. 1. Find the second term in the epansion of ( + ) 6. 15. Find the fourth term in the epansion of (3 ) 6. 16. Use the Binomial Theorem to write the binomial epansion of ( + 5). 17. Use the Binomial Theorem to write the binomial epansion of (a + b) 5. 1. Use the Binomial Theorem to write the binomial epansion of ( 3) 5. 19. Use the Binomial Theorem to write the binomial epansion of ( + y) 3. MATHEMATICAL PRACTICES Make Sense of Problems and Persevere in Solving Them 0. Consider the statement below. Pascal s Triangle In the epansion of every binomial, the powers of decrease by 1 from left to right when written as a polynomial in standard form. a. Epand the binomial ( + 1) 5 and state whether the epansion supports or disproves the statement above and why. b. If the epansion disproves the statement, modify it so that it becomes a true statement. 015 College Board. All rights reserved. 6 SpringBoard Mathematics Algebra, Unit 3 Polynomials
How Many Roots? Write your answers on notebook paper. Show your work. Lesson 17-1 1. State the common factor of the terms in the polynomial 5 3 + 30 10. Then factor the polynomial.. Which of the following is one of the factors of the polynomial 15? A. B. 5 C. 5 + 1 D. 3 1 3. Factor each polynomial. a. 6 + 7 5 b. 1 + 5 + 6. Factor by grouping. a. 3 6 + b. 1 + 3 30 5 5. Factor each difference or sum of cubes. a. 15 3 + 16 b. 6 7 6. Use the formulas for factoring quadratic binomials and trinomials to factor each epression. a. 1 + 33 b. 1 65 c. + 17 + 60 d. 6 100 Lesson 17- ACTIVITY 17 7. Which theorem states that a polynomial of degree n has eactly n linear factors, counting multiple factors? A. Binomial Theorem B. Quadratic Formula C. Fundamental Theorem of Algebra D. Comple Conjugate Root Theorem. Find the zeros of the functions by factoring and using the Zero Product Property. Identify any multiple zeros. a. f () = + 1 b. g() = 3 3 3 c. h() = 5 3 6 5 + 5 d. h() = 3 36 3 + 10 9. The table of values shows coordinate pairs on the graph of f(). Which of the following could be f ()? A. ( + 1)( 1) B. ( 1)( + 1)( 3) C. ( + 1) ( + 3) D. ( + 1)( ) 1 f() 0 0 3 1 0 3 10. Write a polynomial function of nth degree that has the given zeros. a. n = 3; = 1, = 6, = 6 b. n = ; = 3, = 3, = 0, = ACTIVITY 17 Continued ACTIVITY PRACTICE 1. 5; 5( + 6 ). B 3. a. (3 + 5)( 1) b. (7 + )( + 3). a. ( )( + 1) b. (6 + 1)( 3 5) 5. a. (5 + 6)(5 30 + 36) b. ( 3)( + 3 + 9) 6. a. ( 3)( 11) b. (3 + 5)(3 5)(9 + 5) c. ( + 5)( + 1) d. ( 3 + 10)( 3 10) 7. C. a. = ±3i, = 0 (double) b. = 1, = 1± 3 i c. =± 3, = 6 5 d. = 0 (double), = 6 (double) 9. B 10. a. f() = 3 36 + 36 b. f() = 3 9 + 36 017 College Board. All rights reserved. Activity 17 Factors of Polynomials 75
ACTIVITY 17 Continued 11. B 1. a. f() = 3 + 5 50 b. f() = + 16 5 c. f() = 3 + + 10 13. a. n = b. n = 5 c. n = 1. A 15. Sample answer: The polynomial in II is a fifth-degree polynomial; if you multiply the factors in III together, the constant term will not equal. 16. f() = ( ) ( i)( + i); f() = 3 + 17 + 16 ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems. ACTIVITY 17 11. Which of the following polynomial functions has multiple roots at = 0? A. f () = B. f () = 3 C. f () = 3 D. all of the above 1. Write a polynomial function of nth degree that has the given real or comple roots. a. n = 3; =, = 5, = 5 b. n = ; = 3, = 3, = 5i c. n = 3; =, = 1 + i 13. Give the degree of the polynomial function with the given real or comple roots. a. = 7, = 1, = i b. =, =, = 0, = + i c. = i, = 1 3i 1. Which of the following could be the factored form of the polynomial function f () = +... +? I. f () = ( + 1)( + 3)( + i)( i) II. f () = ( + ) ( 1)( + )( 6) III. f () = ( + 3)( )( + i)( i) A. I only B. I and II only C. II only D. I, II, and III How Many Roots? 15. Eplain your reason(s) for eliminating each of the polynomials you did not choose in Item 1. MATHEMATICAL PRACTICES Use Appropriate Tools Strategically 16. Use the information below to write a polynomial function, first in factored form and then in standard form. Fact: The graph only touches the -ais at a double zero; it does not cross through the ais. Clue: One of the factors of the polynomial is ( + i). 100 0 60 0 0 y - 017 College Board. All rights reserved. 76 SpringBoard Mathematics Algebra, Unit 3 Polynomials
Getting to the End Behavior Write your answers on notebook paper. Show your work. Lesson 1-1 For Items 1, match each equation or description to one of the graphs below. 1. an even function with no real roots and a positive leading coefficient. an even function with three real roots and a negative leading coefficient. 3. an odd function with one real root and a negative leading coefficient.. f() = a 3 + b 5. g() = a 3 + + d 6. h() = a + e 7. p() = a 5 + f. p() = a 5 + f For Items 9 11, use what you know about end behavior and zeros to graph each function. 9. f() = + 3 3 + = ( 1)( 6)( + )( + 7) 10. y = 5 1 + 37 3 + 60 155 + 0 = ( 7)( + 5)( ) 3 11. f() = + 11 3 1 59 + 70 = ( 1)( 5)( + )( 7) ACTIVITY 1 1. Make a general statement about what information is revealed by an unfactored polynomial compared to a factored polynomial. 13. Miguel identified the graph below as a polynomial function of the form f( )= a b + c, where a, b, and c are positive real numbers. ACTIVITY 1 Continued ACTIVITY PRACTICE 1. F. C 3. B. E 5. H 6. A 7. G. D 9. y 00 10. 00 00 6000 000 y A. B. 000 C. D. Which reason best describes why Miguel is incorrect? A. The graph is not a fourth-degree polynomial. B. The leading coefficient of Miguel s polynomial should be negative. C. The graph is of an even function, but Miguel s polynomial is not even. D. The y-intercept is below the -ais, so Miguel s polynomial should end with c, not + c. 11. y 100 100 00 6 015 College Board. All rights reserved. E. F. G. H. 1. An unfactored polynomial reveals information about symmetry, end behavior, the number of relative etrema, and the y-intercept of a graph. Factored polynomials reveal information about -intercepts of a graph. 13. D Activity 1 Graphs of Polynomials 9
ACTIVITY 1 Continued 1. a. ± 1, ± 3, ± 1, ± 3 b. ± 1, ± 1, ±, ±, ± 15. y 100 50 50 100 16. a. or 0 possible positive real roots; 1 possible negative real root b. or 0 possible positive real roots; or 0 possible negative real roots 17. y 1. the Comple Conjugate Root Theorem 19. points: (, 10), ( 1, 1), (, 1), (, ); zeros: (1, 0), (, 0) 0. points: ( 1.5, 19.5), ( 1, 1), (1, ), (1.5, 9.5), (3, 300); zeros: ( 3, 0), (0.5, 0) 1. B. < < and > 10 3. < 6 and 1 < < 6. and 5. a. By hand; Sample answer: Since the function is already factored, it will be fairly easy to graph using zeros and end behavior. Also, the scale will be etremely large, so it will take some effort to find a good viewing window in the graphing calculator. b. Graphing calculator; Sample answer: The function is not factorable, and the possible rational roots include fractions. c. By hand; Sample answer: There are enough points in the table to get a good idea of what the function looks like. Also, you don t have an equation to enter into the graphing calculator. ACTIVITY 1 Lesson 1-1. Determine all the possible rational roots of: a. f() = 3 13 6 3 b. g() = + 6 3 3 11 + 15. Graph f() = 3 13 6 3. 16. Determine the possible number of positive and negative real roots for: a. h() = 3 + 5 + b. p() = + 6 3 3 11 + 17. Graph h() = 3 + 5 +. 1. Descartes Rule of Signs states that the number of positive real roots of f() = 0 equals the number of variations in sign of the terms of f(), or is less than this number by an even integer. What theorem offers a reason as to why the number could be less than this number by an even integer? For Items 19 0, apply the Remainder Theorem to all the possible rational roots of the given polynomial to identify points on the graph or zeros of the polynomial. 19. p() = 3 5 + 0. h() = + 5 3 + 5 3 1. The graph of f() has an -intercept at (, 0). Which of the following MUST be true? I. f() = 0 II. is a factor of f(). III. f() also has an -intercept at (, 0). A. II only B. I and II only C. II and III only D. I, II, and III Lesson 1-3 Getting to the End Behavior For Items, solve the polynomial inequality.. ( + )( )( 10) > 0 3. 3 36 + 36 < 0. + 0 3 3 MATHEMATICAL PRACTICES Look For and Epress Regularity in Repeated Reasoning 5. Some polynomial functions are represented in a variety of forms below. For each representation, describe whether you think it is more efficient to graph the polynomial using a graphing calculator or by hand. Justify your choices. a. f() = ( + 15)( + 7)( 5) ( 1) b. g() = + 6 3 3 11 + 7 c. 3 f() 1 1 0 1 1 3 6 015 College Board. All rights reserved. ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems. 90 SpringBoard Mathematics Algebra, Unit 3 Polynomials