ࡱ> y bjbj ~{{L7ppppp$h.44444$!)Qp)pp44zRp4p44pba0*"1Z*"*"p0))4*" 6: MHF4U1-ASSIGNMENT CHAPTER 2 A NAME:___________________________ True/False Indicate whether the statement is true or false. ____ 1. When performing long division of a polynomial by a linear binomial, the degree of the remainder is always smaller than the degree of the divisor. ____ 2. If P(3) = 0 for a polynomial P(x), then x + 3 is a factor of P(x). ____ 3. For a polynomial equation P(x) = 0, if P(x) is not factorable, then P(x) = 0 has no real roots. ____ 4. All quartic polynomial equations have at least one real solution. Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 5. If x3 4x2 + 5x 6 is divided by x 1, then the restriction on x is a.x  4c.x  1b.x  1d.no restrictions ____ 6. What is the remainder when x4 + 2x2 3x + 7 is divided by x + 2? a.25c.37b.13d.9 ____ 7. If 6x4 2x3 21x2 + 7x + 8 is divided by 3x 1 to give a quotient of 2x3 7x and a remainder of 8, then which of the following is true? a.b.6x4 2x3 21x2 + 7x + 8 = (3x 1)(2x3 7x) + 8c.d.all of the above ____ 8. When P(x) = 4x3 4x + 1 is divided by 2x 3, the remainder is a.c. b.P(3) = 97d.  ____ 9. For a polynomial P(x), if P = 0, then which of the following must be a factor of P(x)? a.c.5x + 3b.3x + 5d.5x 3 ____ 10. Which of the following binomials is a factor of x3 6x2 + 11x 6? a.x 1c.x + 7b.x + 1d.2x + 3 ____ 11. Which set of values for x should be tested to determine the possible zeros of x3 2x2 + 3x 12? a.1, 2, 3, 4, 6, and 12c.1, 2, 3, 4, and 6b.1, 2, 3, 4, 6, and 12d.2, 3, 4, 6, and 12 ____ 12. Determine the value of k so that x 3 is a factor of x3 3x2 + x + k. a.k = 3c.k = 1b.k = 3d.k = 1 ____ 13. Find k if 2x + 1 is a factor of kx3 + 7x2 + kx 3. a.k = 2c.k = b.k = 2d.none of the above ____ 14. Which of the following is the fully factored form of x3 + 3x2 x 3? a.(x + 3)(x2 1)c.x2(x + 3) (x + 3)b.(x 1)(x + 1)(x + 3)d.(x2 1)(x 3) ____ 15. Which of the following is the fully factored form of x3 6x2 6x 7? a.(x 7)(x + 1)2c.(x 7)(x2 + x + 1)b.(x 7)(x + 1)(x 1)d.(x 6)(x + 1)(x 1) ____ 16. Which of the following is the factored form of x4 2x2 3? a.(x 1)(x + 1)(x 3)c.(x2 + 1)(x2 3)b.(x2 1)(x + 3)d.none of the above ____ 17. One root of the equation x3 + 2x 3x2 6 = 0 is a.3c.3 b.1d.1  ____ 18. What is the maximum number of real distinct roots that a quartic equation can have? a.infinitely manyc.2b.4d.none of the above ____ 19. If 2 is one root of the equation 4x3 + kx 24 = 0, then the value of k is a.1c.8b.4d.impossible to determine ____ 20. Based on the graph of f(x) = x4 2x3 + 3x + 2 shown, what are the real roots of x4 2x3 + 3x + 2 = 0?  a.2c.impossible to determineb.2, 1, 1, 2d.no real roots ____ 21. Which of the following graphs of polynomial functions corresponds to a cubic polynomial equation with roots 2, 3, and 4? a. c. b. d.  Completion Complete each statement. 22. If P(x) is divided by ax b, then the _______________ is P. 23. If x(4x 3)(x + 1) = 0, then the solutions for x are _______________. 24. The x-intercepts of the graph of a polynomial function correspond to the _______________ of the related polynomial equation. Matching Match the correct term with the correct part of the statement.  a.quotientc.divisorb.remainderd.dividend ____ 25. 4x3 + x 3 ____ 26. 2x 1 ____ 27. 2x2 + x + 1 ____ 28. 2 Short Answer 29. a) Use long division to divide x3 + 3x2 7 by x + 2. Express the result in quotient form. b) Identify any restrictions on the variable. c) Write the corresponding statement that can be used to check the division. d) Verify your answer. 30. Factor fully. a) 29x2 21 + 10x4 b) 2(x + 1)2 32 c) 6x3 7x2 12x + 13 31. Factor fully. a) 2(n 1)2 4(n 1) + 2 b) 2x4 + 7x3 10x2 32x c) x3 x2 x + 1 32. Factor fully. a) x2(x 2)(x + 2) + 3x + 6 b) 16x4 (x + 1)2 c) 2x3 + 5x2 14x 8 33. Solve. a) 3x3 + 2x2 8x + 3 = 0 b) 2x3 + x2 10x 5 = 0 c) 5x4 = 7x2 2 34. Solve by factoring. a) 8x3 36x2 + 46x 15 = 0 b) x4 + 3x2 28 = 0 c) 2x4 54x = 0 Problem 35. The polynomial 6x3 + mx2 + nx 5 has a factor of x + 1. When divided by x 1, the remainder is 4. What are the values of m and n? 36. Factor 2x4 7x3 41x2 53x 21 fully. 37. Show that x + a is a factor of the polynomial P(x) = (x + a)4 + (x + c)4 (a c)4. 38. Given that 2 is a root of x3 + x = 4x2 + 6, find the other root(s). 39. The height of a square-based box is 4 cm more than the side length of its square base. If the volume of the box is 225 cm3, what are its dimensions? 40. Amit has designed a rectangular storage unit to hold large factory equipment. His scale model has dimensions 1 m by 2 m by 4 m. By what amount should he increase each dimension to produce an actual storage unit that is 9 times the volume of his scale model? MPJNS-MHF4U1-ASSIGNMENT CHAPTER 2 A Answer Section TRUE/FALSE 1. ANS: T PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: remainder 2. ANS: T PTS: 1 DIF: 1 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: factor theorem 3. ANS: F PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots 4. ANS: F PTS: 1 DIF: 1 REF: Knowledge and Understanding; Thinking OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots MULTIPLE CHOICE 5. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: restriction 6. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: remainder theorem 7. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: quotient, remainder 8. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: remainder theorem 9. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: factor theorem 10. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding; Application OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem, integral zero theorem 11. ANS: B PTS: 1 DIF: 1 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: integral zero theorem 12. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding; Application OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem 13. ANS: A PTS: 1 DIF: 3 REF: Knowledge and Understanding; Application OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem 14. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factored form 15. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factored form 16. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factored form 17. ANS: C PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots 18. ANS: B PTS: 1 DIF: 1 REF: Knowledge and Understanding; Thinking OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots 19. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding; Application OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots 20. ANS: D PTS: 1 DIF: 1 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots 21. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots COMPLETION 22. ANS: remainder PTS: 1 DIF: 1 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: remainder theorem 23. ANS:  PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions KEY: polynomial equation 24. ANS: roots solutions PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions KEY: x-intercepts, polynomial equation MATCHING 25. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: quotient, remainder, divisor, dividend 26. ANS: C PTS: 1 27. ANS: A PTS: 1 28. ANS: B PTS: 1 SHORT ANSWER 29. ANS: a)  b) x  2 c) x3 + 3x2 7 = (x + 2)(x2 + x 2) 3 d) Expand to verify. PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions KEY: long division, restriction 30. ANS: a) (2x2 + 7)(5x2 3) b) 2(x 3)(x + 5) c) (x 1)(6x2 x 13) PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor polynomial expression NOT: A variety of factoring techniques is required. 31. ANS: a) 2(n 2)2 b) x(x + 2)(2x2 + 3x 16) c) (x + 1)(x 1)2 PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor polynomial expression NOT: A variety of factoring techniques is required. 32. ANS: a) (x + 2)(x + 1)(x2 3x + 3) b) (4x2 x 1)(4x2 + x + 1) c) (2x + 1)(x 2)(x + 4) PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor polynomial expression NOT: A variety of factoring techniques is required. 33. ANS: a)  b)  c)  PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions KEY: polynomial equation NOT: A variety of factoring techniques is required. 34. ANS: a)  b) 2, 2 c) 0, 3 PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions KEY: polynomial equation NOT: A variety of factoring techniques is required. PROBLEM 35. ANS: Let P(x) = 6x3 +mx2 + nx 5. By the factor theorem, since x + 1 is a factor of P(x), then P(1) = 0.  By the remainder theorem, when P(x) is divided by x 1, the remainder is P(1). Solve P(1) = 4.  Solve the system of equations.  Substitute m = 3 into equation (2), to find that n = 8. PTS: 1 DIF: 4 REF: Knowledge and Understanding; Application; Thinking OBJ: Sections 2.1, 2.2 LOC: C3.1, C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem, remainder theorem 36. ANS: Let P(x) = 2x4  7x3  41x2  53x  21. By the rational zero theorem, possible values of  are 1, 3, 7, 21, , and . Test the values to find a zero. Since x =  1 is a zero of P(x), x + 1 is a factor. Divide to determine the other factor. 2x4 7x3 41x2 53x 21 = (x + 1)(2x3 9x2 32x 21) Factor 2x3 9x2 32x 21 using a similar method. 2x3 9x2 32x 21 = (x + 1)(2x2 11x 21) = (x + 1)(2x + 3)(x 7) So, P(x) = 2x4 7x3 41x2 53x 21 = (x + 1)2(2x + 3)(x 7). PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem, rational zero theorem 37. ANS: By the factor theorem, x + a is a factor of P(x) if P(a) = 0.  PTS: 1 DIF: 2 REF: Knowledge and Understanding; Thinking OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem 38. ANS: Rewrite in the form P(x) = 0. x3 + 4x2 + x 6 = 0 Since 2 is a root, x + 2 is a factor of P(x). Divide to find the other factor. x3 + 4x2 + x 6 = (x + 2)(x2 + 2x 3) = (x + 2)(x + 3)(x 1) Solve x3 + 4x2 + x 6 = 0. (x + 2)(x + 3)(x 1) = 0 x + 2 = 0 or x + 3 = 0 or x 1 = 0 x = 2 or x = 3 or x = 1 Thus, the other roots are 3 and 1. PTS: 1 DIF: 3 REF: Knowledge and Understanding; Application OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions KEY: real roots, polynomial equation 39. ANS: Let x represent the side length of the base. Then, V(x) = x2(x + 4). Solve x2(x + 4) = 225. x2(x + 4) 225 = 0 x3 + 4x2 225 = 0 Factor the corresponding polynomial function. Use the integral zero theorem to determine the possible values of b are 1, 3, 5, 9, 15, 25, 45, 75, and 225. Test only positive values since x represents a side length. Since x = 5 is a zero of the function, x  5 is a factor. Divide to determine the other factor. x3 + 4x2  225 = (x  5)(x2 + 9x + 45) Solve (x  5)(x2 + 9x + 45) = 0. x 5 = 0 or x2 + 9x + 45 = 0 x = 5 or no solution The dimensions of the box are 5 cm by 5 cm by 9 cm. PTS: 1 DIF: 3 REF: Knowledge and Understanding; Thinking; Application OBJ: Section 2.3 LOC: C3.7 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots 40. ANS: Let x represent the increase in each dimension. Then V(x) = (1 + x)(2 + x)(4 + x). 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<<#h[AB*CJOJQJ^JaJphh[AB*CJaJphh[A6B*CJ]aJphh[AB*CJaJphh[AB*CJH*aJphI_:::::;.;B;U;<==>0>E>y>z>*$1$7$8$H$gd[A$ *$7$8$H$gd[A $*$7$8$H$gd[A  Lz Ll*$1$7$8$H$`gd[A *$1$7$8$H$gd[A z Ll*$1$7$8$H$gd[A<<<< <"<*<,<4<6<><@<P<R<<<<<"=$=================>>>>>>> >!>%>&>0>1>x>y>&?(?7?8?h?i?j?k?t?u?{?|???????????h[AB*CJaJphh[AB*CJH*aJphh[A6B*CJ]aJphh[AB*CJaJph#h[AB*CJOJQJ^JaJphHz>>?'?(?3?????Gs9{*$1$7$8$H$gd[A $*$7$8$H$gd[A  Lz Ll*$1$7$8$H$`gd[A *$1$7$8$H$gd[A z Ll*$1$7$8$H$gd[A????????????????GHIMNOTU^_efgklz{h!Lh[AB*CJaJphUh[A6B*CJ]aJphh[AB*CJaJphh[AB*CJH*aJph3Factor the corresponding polynomial function using the factor theorem. x3 + 7x2 + 14x 64 = (x 2)(x2 + 9x + 32) Solve (x 2)(x2 + 9x + 32) = 0. x 2 = 0 or x2 + 9x + 32 = 0 x = 2 or no solution Each dimension should be increased by 2 m. 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