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3.6e: Exercises - Zeroes of Polynomial Functions

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A: Concepts

Exercise 3.6e.A: Concepts

1) Describe a use for the Remainder Theorem.

2) Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.

3) What is the difference between rational and real zeros?

4) If Descartes’ Rule of Signs reveals a 0 or 1 change of signs, what specific conclusion can be drawn?

5) If synthetic division reveals a zero, why should we try that value again as a possible solution?

Answers to odd exercises: 

1. The theorem can be used to evaluate a polynomial.

3. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

5. Polynomials can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

B: Use the Remainder Theorem to Evaluate a Polynomial

Exercise 3.6e.B: Use the Remainder Theorem

 Use synthetic division to evaluate p(c) and write p(x) in the form p(x)=(xc)q(x)+r.

6. p(x)=2x2x+1, c=4

7. p(x)=4x233x180, c=12

8. p(x)=2x3x+6, c=3

9. p(x)=x3+2x2+3x+4, c=1

10. p(x)=3x36x2+4x8, c=2

11. p(x)=8x3+12x2+6x+1, c=12

12. p(x)=2x4+x34x2+10x7, c=32

13.  p(x)=x43x320x224x8, c=7 

14. p(x)=x45x28x12, c=3

15. p(x)=x45x3+x2+5, c=2

Answers to odd exercises:

7. p(12)=0, p(x)=(x12)(4x+15)

9. p(1)=2p(x)=(x+1)(x2+x+2)+2

11. p(12)=0, p(x)=(2x+1)(4x2+4x+1)

13. p(7)=216p(x)=(x7)(x3+4x2+8x+32)+216

15. p(2)=15p(x)=(x2)(x33x25x10)15

C: Given one zero or factor, find all Real Zeros, and factor a polynomial

Exercise 3.6e.C: Use the Factor Theorem given one zero or factor

 Given a polynomial and one of its factors, find the rest of the real zeros and write the polynomial as a product of linear and irreducible quadratic factors.

17) f(x)=2x3+x25x+2; Factor: (x+2)

18) f(x)=3x3+x220x+12; Factor: (x+3)

19) f(x)=2x3+3x2+x+6; Factor: (x+2)

20) f(x)=5x3+16x29; Factor: (x3)

21) f(x)=x3+3x2+4x+12; Factor: (x+3)

22) f(x)=4x37x+3; Factor: (x1)

23) f(x)=2x3+5x212x30; Factor: (2x+5)

24) f(x)=2x39x2+13x6; Factor: (x1)

Answers to odd exercises:

17. 2,1,12; f(x)=(x+2)(x1)(2x1)

19. 2; f(x)=(x+2)(2x2x+3)

21. 3; f(x)=(x+3)(x2+4)

23. 52,6,6; f(x)=(2x+5)(x6)(x+6)

 Given a polynomial and c, one of its zeros, find the rest of the real zeros and write the polynomial as a product of linear and irreducible quadratic factors. It is possible some factors are repeated.

25. p(x)=x324x2+192x512,c=8

26. p(x)=3x3+4x2x2,c=23

27. p(x)=2x33x211x+6,c=12

28. p(x)=x3+2x23x6,c=2

29. p(x)=2x3x210x+5,c=12

30. p(x)=4x428x3+61x242x+9,c=12 

31. p(x)=x5+2x412x338x237x12, c=1 

32. p(x)=2x5+7x418x28x+8,  c=12

33. p(x)=3x5+2x415x310x2+12x+8,  c=23

Answers to odd exercises:

25. zeros: 8p(x)=(x8)3

27. zeros: 12,2,3p(x)=(2x1)(x+2)(x3)

29. zeros: 12,±5p(x)=(2x1)(x+5)(x5)

31. zeros: 1, 3, 4p(x)=(x+1)3(x+3)(x4)

33. zeros: 2,1,23,1,2;
  p(x)=(x+2)(x+1)(x1)(x2)(3x+2)

D: List all Possible Rational Zeros

Exercise 3.6e.D: Use the Rational Zero Theorem

 Use the Rational Zeros Theorem to list all possible rational zeros for each given function. 

35) f(x)=2x3+3x28x+5

36) f(x)=3x3+5x25x+4

37) f(x)=6x410x2+13x+1

38) f(x)=4x510x4+8x3+x28

39. f(x)=x32x25x+6

40. f(x)=x4+2x312x240x32

41. f(x)=x37x2+x7

42. f(x)=x3+4x211x+6

43. f(x)=x49x24x+12

44. f(x)=2x3+19x249x+20

45. f(x)=17x3+5x2+34x10

46. f(x)=36x412x311x2+2x+1

47. f(x)=3x3+3x211x10

48. f(x)=2x4+x37x23x+3

Answers to odd exercises:

35. ±5,±1,±12,±52

37. ±1,±12,±13,±16

39. ±1, ±2, ±3, ±6  41. ±1, ±7

43. ±1, ±2, ±3, ±4, ±6, ±12

45. ±1, ±2, ±5, ±10, ±117,±217,±517,±1017

47. ±1, ±2, ±5, ±10, ±13,±23,±53,±103

E: Find all Zeros that are Rational

Exercise 3.6e.E: Find all zeros that are rational

 Use the Rational Zero Theorem to find all real number zeros.

49) x33x210x+24=0

50) 2x3+7x210x24=0

51) x3+2x29x18=0

52) x3+5x216x80=0

53) x33x225x+75=0

54) 2x33x232x15=0

55) 2x3+x27x6=0

56) 2x33x2x+1=0

57) 3x3x211x6=0

58) x42x37x2+8x+12=0

59) 4x33x+1=0

60) 4x4+4x325x2x+6=0

61) x4+2x39x22x+8=0

62) x4+2x34x210x5=0

63) 5x4+4x319x2+16x+4=0

Answers to odd exercises:
49. 3,2,4

51. 2,3,3

53. 3,5,5

55. 2,1,32

57. 23,1±132

59. 1,12

61. 1,2,1,4

63. 15,1

 Find the real zeros of the polynomial. State the multiplicity of each real zero.

65. f(x)=x32x25x+6

66. f(x)=x3+4x211x+6

67. f(x)=x49x24x+12

68. f(x)=17x3+5x2+34x10

69. f(x)=36x412x311x2+2x+1

70. f(x)=2x4+x37x23x+3

71. f(x)=2x3+7x2+4x4

72. f(x)=2x43x3+10x2+12x8

Answers to odd exercises:

65. x=1 (mult. 1), x=3 (mult. 1), x=2 (mult. 1)

67. x=2 (mult. 2), x=1 (mult. 1), x=3 (mult. 1)

69. x=12 (mult. 2), x=13 (mult. 2)

71.  x=2 (mult. 2), x=12 (mult. 1)

F: Find all zeros (both real and imaginary)

Exercise 3.6e.F: Find all zeros

 Use the Rational Zero Theorem to find all complex solutions (real and non-real).

72.5) x3+x2+x+1=0

73) x38x2+25x26=0

74) x3+13x2+57x+85=0

75) 3x34x2+11x+10=0

76) x4+2x3+22x2+50x75=0

77) 2x33x2+32x+17=0

Answers to odd exercises:
73. 2,3+2i,32i 75. 23,1+2i,12i 77. 12,1+4i,14i

G: Find all zeros and sketch

Exercise 3.6e.G: Find all zeros and sketch

 Determine the end behaviour, all the real zeros, their multiplicity, and y-intercept. Sketch the function. (Use synthetic division to find a rational zero. Use the quotient to find the next zero).

78) f(x)=x31

79) f(x)=x4x21

80) f(x)=x32x25x+6

81) f(x)=2x3+37x2+200x+300 

82) f(x)=x4+2x312x2+14x5

83) f(x)=2x45x35x2+5x+3

84) f(x)=x32x216x+32

85) f(x)=x37x28x+16

86. f(x)=x44x3+3x2+10x8

87. f(x)=x46x3+8x2+6x9

88. f(x)=x4+4x35x236x36

89. f(x)=x5x45x3+x2+8x+4

90. f(x)=x4+2x3+6x9

Answers to odd exercises:

79. zeros (odd multiplicity): ±1+52, 2 imaginary zeros, y-intercept (0,1)

      CNX_PreCalc_Figure_03_06_202.jpg

81. zeros (odd multiplicity): {10,6,52}; y-intercept: (0,300)

  CNX_PreCalc_Figure_03_06_206 (1).jpg

83. zeros (odd multiplicity); {1,1,3,12}, y-intercept (0,3)

      CNX_PreCalc_Figure_03_06_208.jpg

85. zeros; 4 (multiplicity 2), 1 (multiplicity 1), y-intercept (0,16).   

3.6E #85.png

87.  odd multiplicity zeros: {1,1}; even multiplicity zero: {3}; y-intercept (0,9)

       3.6E #87.png

89. odd multiplicity zero: {1}, even multiplicity zero {2}. y-intercept (0,4)

      3.6E #89.png

H: Given zeros, construct a polynomial function

Exercise 3.6e.H: Given zeros, construct a polynomial function

 Construct a polynomial function of least degree possible using the given information. You may leave the polynomial in factored form.

91) A lowest degree polynomial with real coefficients and zero 3i 

92) A lowest degree polynomial with rational coefficients and zeros: 2 and 6

93) A lowest degree polynomial with integer coefficients and Real roots: 1 (with multiplicity 2), and 1.

94) A lowest degree polynomial with integer coefficients and Real roots: 2, and 12 (with multiplicity 2

95) A lowest degree polynomial with integer coefficients and Real roots:12,0,12 

96) A lowest degree polynomial with integer coefficients and Real roots: 4,1,1,4 

97) A lowest degree polynomial with integer coefficients and Real roots: 1,1,3 

98. A lowest degree polynomial with real coefficients and zeros: 2 and 5i

99. A lowest degree polynomial with real coefficients and zeros: 4 and 2i.

100. The solutions to p(x)=0 are x=±3 and x=6.  The leading term of p(x) is 7x4
The point (3,0) is a local minimum on the graph of y=p(x).

101. The solutions to p(x)=0 are x=±3, x=2, and x=4, The leading term of p(x) is x5
The point (2,0) is a local maximum on the graph of y=p(x).

102. p is degree 4. as x, p(x) p has exactly three x-intercepts: (6,0), (1,0) and (117,0)
The graph of y=p(x) crosses through the x-axis at (1,0).

103. Find a quadratic polynomial with integer coefficients which has x=35±295 as its real zeros.

Answers to odd exercises:

91. f(x)=(x2+9)

93. f(x)=(x+1)2(x1)

95. f(x)=x(2x+1)(2x1)

97. f(x)=(x+1)(x1)(x3)

99. p(x)=(x4)(x2i)(x+2i)=x34x2+4x16

101. p(x)=(x+2)2(x3)(x+3)(x4)

103. p(x)=5x26x4

I: Use Intermediate Value Theorem

Exercise 3.6e.I: Intermediate Value Theorem

 Use the Intermediate Value Theorem to confirm the polynomial f has at least one zero within the given interval.

104) f(x)=x39x, between x=4 and x=2.

105) f(x)=x39x, between x=2 and x=4.

106) f(x)=x52x, between x=1 and x=2.

107) f(x)=x4+4, between x=1 and x=3.

108) f(x)=2x3x, between x=1 and x=1.

109) f(x)=x3100x+2, between x=0.01 and x=0.1

Answers to odd exercises:
105. f(2)=10,f(4)=28.
Sign change confirms.
107. f(1)=3,f(3)=77.
Sign change confirms.
109. f(0.01)=1.000001,f(0.1)=7.999.
Sign change confirms.

x


3.6e: Exercises - Zeroes of Polynomial Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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