College Algebra, 2013a

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266 Polynomial Functions In Exercises 31 - 40, you are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial. 31. x 3 − 6x 2 +11x − 6, c = 1 32. x 3 − 24x 2 + 192x − 512, c =8 33. 3x 3 +4x 2 − x − 2, c = 2 3 34. 2x 3 − 3x 2 − 11x +6, c = 1 2 35. x 3 +2x 2 − 3x − 6, c = −2 36. 2x 3 − x 2 − 10x +5, c = 1 2 37. 4x 4 − 28x 3 +61x 2 − 42x +9,c = 1 2 is a zero of multiplicity 2 38. x 5 +2x 4 − 12x 3 − 38x 2 − 37x − 12, c = −1 is a zero of multiplicity 3 39. 125x 5 − 275x 4 − 2265x 3 − 3213x 2 − 1728x − 324, c = − 3 5 is a zero of multiplicity 3 40. x 2 − 2x − 2, c =1− √ 3 In Exercises 41 - 45, create a polynomial p which has the desired characteristics. You may leave the polynomial in factored form. 41. ˆ The zeros of p are c = ±2 andc = ±1 ˆ The leading term of p(x) is 117x 4 . 42. ˆ The zeros of p are c = 1 and c =3 ˆ c = 3 is a zero of multiplicity 2. ˆ The leading term of p(x) is−5x 3 43. ˆ The solutions to p(x) =0arex = ±3 andx =6 ˆ The leading term of p(x) is7x 4 ˆ The point (−3, 0) is a local minimum on the graph of y = p(x). 44. ˆ The solutions to p(x) =0arex = ±3, x = −2, and x =4. ˆ The leading term of p(x) is−x 5 . ˆ The point (−2, 0) is a local maximum on the graph of y = p(x). 45. ˆ p is degree 4. ˆ as x →∞, p(x) →−∞ ˆ p has exactly three x-intercepts: (−6, 0), (1, 0) and (117, 0) ˆ The graph of y = p(x) crosses through the x-axis at (1, 0). 46. Find a quadratic polynomial with integer coefficients which has x = 3 5 ± √ 29 5 as its real zeros.
3.2 The Factor Theorem and the Remainder Theorem 267 3.2.2 Answers 1. 4x 2 +3x − 1=(x − 3)(4x + 15) + 44 2. 2x 3 − x +1= ( x 2 + x +1 ) (2x − 2) + (−x +3) 3. 5x 4 − 3x 3 +2x 2 − 1= ( x 2 +4 )( 5x 2 − 3x − 18 ) +(12x + 71) 4. −x 5 +7x 3 − x = ( x 3 − x 2 +1 )( −x 2 − x +6 ) + ( 7x 2 − 6 ) 5. 9x 3 +5=(2x − 3) ( 9 2 x2 + 27 4 x + 81 8 ) + 283 8 6. 4x 2 − x − 23 = ( x 2 − 1 ) (4) + (−x − 19) 7. ( 3x 2 − 2x +1 ) =(x − 1) (3x +1)+2 8. ( x 2 − 5 ) =(x − 5) (x +5)+20 9. ( 3 − 4x − 2x 2) =(x +1)(−2x − 2) + 5 10. ( 4x 2 − 5x +3 ) =(x +3)(4x − 17)+54 11. ( x 3 +8 ) =(x +2) ( x 2 − 2x +4 ) +0 12. ( 4x 3 +2x − 3 ) =(x − 3) ( 4x 2 +12x +38 ) + 111 13. ( 18x 2 − 15x − 25 ) = ( x − 5 3) (18x + 15) + 0 14. ( 4x 2 − 1 ) = ( x − 1 2) (4x +2)+0 15. ( 2x 3 + x 2 +2x +1 ) = ( x + 1 2)( 2x 2 +2 ) +0 16. ( 3x 3 − x +4 ) = ( x − 2 3)( 3x 2 +2x + 1 3) + 38 9 17. ( 2x 3 − 3x +1 ) = ( x − 1 2)( 2x 2 + x − 5 2) − 1 4 18. ( 4x 4 − 12x 3 +13x 2 − 12x +9 ) = ( x − 3 2)( 4x 3 − 6x 2 +4x − 6 ) +0 19. ( x 4 − 6x 2 +9 ) = ( x − √ 3 )( x 3 + √ 3 x 2 − 3x − 3 √ 3 ) +0 20. ( x 6 − 6x 4 +12x 2 − 8 ) = ( x + √ 2 )( x 5 − √ 2 x 4 − 4x 3 +4 √ 2 x 2 +4x − 4 √ 2 ) +0 21. p(4) = 29 22. p(12) = 0, p(x) =(x − 12)(4x + 15) 23. p(−3) = −45 24. p(−1) = 2 25. p(2) = 0, p(x) =(x − 2) ( 3x 2 +4 ) 26. p ( − 1 ( )( 2) =0,p(x) = x + 1 2 8x 2 +8x +2 )
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College Algebra Version ⌊π⌋ Co
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Table of Contents Preface ix 1 Rela
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Table of Contents v 4.3.3 Answers .
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Preface Thank you for your interest
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xi text and we have gone to great l
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Chapter 1 Relations and Functions 1
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1.1 Sets of Real Numbers and the Ca
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1.2 Relations 21 Solution. 1. To gr
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1.2 Relations 23 lines y = 1 and y
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1.2 Relations 25 ‘connecting the
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1.2 Relations 27 (x − 2) 2 + y 2
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1.2 Relations 29 1.2.2 Exercises In
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1.2 Relations 31 29. 5 4 3 y 30. 2
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1.2 Relations 33 1.2.3 Answers 1. y
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1.2 Relations 35 13. y 14. y 3 2
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1.2 Relations 37 41. y = x 2 +1 The
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1.2 Relations 39 45. y = √ x −
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1.2 Relations 41 49. (x +2) 2 + y 2
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1.3 Introduction to Functions 43 1.
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1.3 Introduction to Functions 45 So
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1.3 Introduction to Functions 47 Al
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1.3 Introduction to Functions 51 23
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1.4 Function Notation 55 1.4 Functi
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1.4 Function Notation 57 (c) To fin
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1.4 Function Notation 59 1 − 4x x
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1.4 Function Notation 61 these limi
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1.4 Function Notation 63 1.4.2 Exer
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1.4 Function Notation 65 ⎧ ⎪⎨
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1.4 Function Notation 67 (b) How mu
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1.4 Function Notation 69 1.4.3 Answ
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1.4 Function Notation 71 18. For f(
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1.4 Function Notation 73 26. For f(
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1.4 Function Notation 75 69. C(0) =
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1.5 Function Arithmetic 77 ( ) 4. F
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1.5 Function Arithmetic 79 Next, we
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1.5 Function Arithmetic 81 r(x + h)
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1.5 Function Arithmetic 83 Solution
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1.5 Function Arithmetic 85 27. f(x)
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1.5 Function Arithmetic 87 1.5.2 An
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1.5 Function Arithmetic 89 13. For
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1.5 Function Arithmetic 91 37. 39.
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1.6 Graphs of Functions 93 1.6 Grap
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1.6 Graphs of Functions 95 In the p
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1.6 Graphs of Functions 97 The calc
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1.6 Graphs of Functions 99 The calc
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1.6 Graphs of Functions 101 on [−
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1.6 Graphs of Functions 103 1. Find
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1.6 Graphs of Functions 105 Example
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1.6 Graphs of Functions 107 1.6.2 E
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1.6 Graphs of Functions 109 In Exer
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1.6 Graphs of Functions 111 For Exe
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1.6 Graphs of Functions 113 y y 4 5
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1.6 Graphs of Functions 115 6. f(x)
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1.6 Graphs of Functions 117 17. y 1
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1.6 Graphs of Functions 119 90. h i
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1.7 Transformations 121 7 y 7 y (5,
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1.7 Transformations 123 2 to all of
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1.7 Transformations 125 2 1 (0, 0)
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1.7 Transformations 127 Solution. 1
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1.7 Transformations 129 y y 3 2 1 (
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1.7 Transformations 131 A few remar
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1.7 Transformations 133 6 y 6 y (4,
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1.7 Transformations 135 Finally, to
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1.7 Transformations 137 (a, f(a)) a
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1.7 Transformations 139 shift right
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1.7 Transformations 141 Thecomplete
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1.7 Transformations 143 64. The gra
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1.7 Transformations 145 25. y =2−
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1.7 Transformations 147 38. g(x) =f
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1.7 Transformations 149 50. y = S 1
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Chapter 2 Linear and Quadratic Func
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2.1 Linear Functions 153 P 2 5. m =
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2.1 Linear Functions 155 Equation 2
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2.1 Linear Functions 157 y 4 3 2 y
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2.1 Linear Functions 159 4. We find
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2.1 Linear Functions 161 average sp
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2.1 Linear Functions 163 2.1.1 Exer
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2.1 Linear Functions 165 40. A rest
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2.1 Linear Functions 167 61. y = 2
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2.1 Linear Functions 169 2.1.2 Answ
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2.1 Linear Functions 171 35. (a) F
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2.2 Absolute Value Functions 173 2.
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2.2 Absolute Value Functions 177 By
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2.2 Absolute Value Functions 179 Ex
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2.2 Absolute Value Functions 181 wh
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2.2 Absolute Value Functions 185 24
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2.2 Absolute Value Functions 187 31
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2.3 Quadratic Functions 189 y y (
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2.3 Quadratic Functions 191 shows t
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2.3 Quadratic Functions 193 From g(
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2.3 Quadratic Functions 195 ( a x +
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2.3 Quadratic Functions 197 To find
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2.3 Quadratic Functions 199 Neverth
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2.3 Quadratic Functions 201 15. The
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2.3 Quadratic Functions 203 2.3.2 A
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2.3 Quadratic Functions 205 8. f(x)
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2.3 Quadratic Functions 207 25. (a)
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2.4 Inequalities with Absolute Valu
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Page 247 and 248: 3.1 Graphs of Polynomials 237 of th
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4.1 Introduction to Rational Functi
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4.1 Introduction to Rational Functi
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4.2 Graphs of Rational Functions 32
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4.3 Rational Inequalities and Appli
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Chapter 5 Further Topics in Functio
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5.1 Function Composition 361 2. As
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5.1 Function Composition 363 (+)
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5.1 Function Composition 365 9. The
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5.1 Function Composition 367 Theore
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5.1 Function Composition 369 5.1.1
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5.1 Function Composition 371 50. (g
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5.1 Function Composition 373 7. For
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5.1 Function Composition 375 20. Fo
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5.1 Function Composition 377 56. (g
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5.2 Inverse Functions 379 The main
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5.2 Inverse Functions 381 y = f −
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5.2 Inverse Functions 383 (b) We ca
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5.2 Inverse Functions 385 y = f(x)
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5.2 Inverse Functions 387 = = 2x 2x
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5.2 Inverse Functions 389 We get g
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5.2 Inverse Functions 391 Using wha
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5.2 Inverse Functions 393 1. Explai
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5.2 Inverse Functions 395 26. Show
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5.3 Other Algebraic Functions 397 5
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5.3 Other Algebraic Functions 399 I
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5.3 Other Algebraic Functions 401 b
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5.3 Other Algebraic Functions 405
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5.3 Other Algebraic Functions 415 (
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Chapter 6 Exponential and Logarithm
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6.1 Introduction to Exponential and
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6.2 Properties of Logarithms 437 6.
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6.2 Properties of Logarithms 439 lo
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6.2 Properties of Logarithms 441 ex
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6.2 Properties of Logarithms 443 as
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6.3 Exponential Equations and Inequ
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6.4 Logarithmic Equations and Inequ
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6.5 Applications of Exponential and
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Chapter 7 Hooked on Conics 7.1 Intr
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7.1 Introduction to Conics 497 If t
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7.2 Circles 499 4 y 3 2 1 −4 −3
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7.2 Circles 501 We close this secti
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7.2 Circles 503 7.2.2 Answers 1. (x
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7.3 Parabolas 505 7.3 Parabolas We
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7.3 Parabolas 507 The distance from
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7.3 Parabolas 509 Example 7.3.3. Gr
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7.3 Parabolas 511 Every cross secti
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7.3 Parabolas 513 7.3.2 Answers 1.
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7.3 Parabolas 515 8. ( y + 2) 3 2 (
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7.4 Ellipses 517 Minor Axis Major A
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7.4 Ellipses 519 This equation is f
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7.4 Ellipses 521 As with circles an
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7.4 Ellipses 523 From this sketch,
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7.4 Ellipses 525 7.4.1 Exercises In
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7.4 Ellipses 527 7.4.2 Answers x 2
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7.4 Ellipses 529 (x − 4) 2 (y −
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7.5 Hyperbolas 531 7.5 Hyperbolas I
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7.5 Hyperbolas 533 endpoints of the
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7.5 Hyperbolas 535 from the center
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7.5 Hyperbolas 537 As with the othe
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7.5 Hyperbolas 539 y 6 5 4 3 2 Jeff
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7.5 Hyperbolas 541 7.5.1 Exercises
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7.5 Hyperbolas 543 is a hyperbola.
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7.5 Hyperbolas 545 (y − 3) 2 (x
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7.5 Hyperbolas 547 19. (x − 1) 2
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Chapter 8 Systems of Equations and
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8.1 Systems of Linear Equations: Ga
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8.2 Systems of Linear Equations: Au
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8.3 Matrix Arithmetic 579 ⎡ ⎤
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8.3 Matrix Arithmetic 581 As did ma
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8.3 Matrix Arithmetic 583 While the
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8.3 Matrix Arithmetic 585 Using Ri
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8.3 Matrix Arithmetic 587 C 2 − 5
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8.3 Matrix Arithmetic 589 ( √2 2
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8.3 Matrix Arithmetic 591 8.3.1 Exe
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8.3 Matrix Arithmetic 593 28. Now l
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8.3 Matrix Arithmetic 595 8.3.2 Ans
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8.3 Matrix Arithmetic 597 [ ] −9
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8.4 Systems of Linear Equations: Ma
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8.5 Determinants and Cramer’s Rul
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8.6 Partial Fraction Decomposition
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8.7 Systems of Non-Linear Equations
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20. Solve the following system ⎧
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Chapter 9 Sequences and the Binomia
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9.1 Sequences 653 3. From {2n − 1
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9.1 Sequences 655 Solution. A good
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9.1 Sequences 657 Looking at the de
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9.1 Sequences 659 28. 0.9, 0.99, 0.
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9.2 Summation Notation 661 9.2 Summ
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9.2 Summation Notation 663 5∑ (
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9.2 Summation Notation 665 S = n (2
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9.2 Summation Notation 667 payment
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9.2 Summation Notation 669 n∑ k=1
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9.2 Summation Notation 671 In Exerc
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9.3 Mathematical Induction 673 9.3
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9.3 Mathematical Induction 675 This
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9.3 Mathematical Induction 677 it i
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9.3 Mathematical Induction 679 9.3.
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9.4 The Binomial Theorem 681 9.4 Th
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9.4 The Binomial Theorem 683 songs
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9.4 The Binomial Theorem 685 (a + b
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9.4 The Binomial Theorem 687 ∑k
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9.4 The Binomial Theorem 689 ( 0 0)
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9.4 The Binomial Theorem 691 9.4.1
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Index n th root of a complex number
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Index 1071 central angle, 701 chang
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Index 1073 reflective property, 523
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Index 1075 matrix, multiplicative,
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Index 1077 midpoint definition of,
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Index 1079 instantaneous, 161, 472
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Index 1081 inconsistent, 553 indepe
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College Algebra, 2013a

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