CHAPTER 2 Polynomial and Rational Functions Section 2.1 Quadratic Functions and Models . 136 Section 2.2 Polynomial Functions of Higher Degree . 151 Section 2.3 Polynomial and Synthetic Division . 168 Section 2.4 Complex Numbers . 180 Section 2.5 Zeros of Polynomial Functions . 187 Section 2.6 Rational Functions . 205 Section 2.7 Nonlinear Inequalities . 222 Review Exercises . 237 Problem Solving . 258 Practice Test . 262 CHAPTER 2 Polynomial and Rational Functions Section 2.1 Quadratic Functions and Models You should know the following facts about parabolas. ■ f ͑x͒ ϭ ax2 ϩ bx ϩ c, a 0, is a quadratic function, and its graph is a parabola. ■ If a > 0, the parabola opens upward and the vertex is the point with the minimum y-value. If a < 0, the parabola opens downward and the vertex is the point with the maximum y-value. ■ The vertex is ͑Ϫb͞2a, f ͑Ϫb͞2a͒͒. ■ To find the x-intercepts (if any), solve ax2 ϩ bx ϩ c ϭ 0. ■ The standard form of the equation of a parabola is f ͑x͒ ϭ a͑x Ϫ h͒2 ϩ k where a 0. (a) The vertex is ͑h, k͒. (b) The axis is the vertical line x ϭ h. Vocabulary Check 1. nonnegative integer; real 2. quadratic; parabola 3. axis or axis of symmetry 4. positive; minimum 5. negative; maximum 1. f ͑x͒ ϭ ͑x Ϫ 2͒2 opens upward and has vertex ͑2, 0͒. 2. f ͑x͒ ϭ ͑x ϩ 4͒2 opens upward and has vertex ͑Ϫ4, 0͒. Matches graph (g). Matches graph (c). 3. f ͑x͒ ϭ x2 Ϫ 2 opens upward and has vertex ͑0, Ϫ2͒. 4. f ͑x͒ ϭ 3 Ϫ x2 opens downward and has vertex ͑0, 3͒. Matches graph (b). Matches graph (h). 5. f ͑x͒ ϭ 4 Ϫ ͑x Ϫ 2͒2 ϭϪ͑x Ϫ 2͒2 ϩ 4 opens downward 6. f ͑x͒ ϭ ͑x ϩ 1͒2 Ϫ 2 opens upward and has vertex and has vertex ͑2, 4͒. Matches graph (f). ͑Ϫ1, Ϫ2͒. Matches graph (a). 7. f ͑x͒ ϭϪ͑x Ϫ 3͒2 Ϫ 2 opens downward and has 8. f ͑x͒ ϭϪ͑x Ϫ 4͒2 opens downward and has vertex ͑4, 0͒. vertex ͑3, Ϫ2͒. Matches graph (e). Matches graph (d). 136 Section 2.1 Quadratic Functions and Models 137 ϭ 1 2 ϭϪ1 2 9. (a) y 2x (b) y 8x y y 5 6 4 4 3 2 2 x −6 −4 46 1 −2 x −4 −3 −2 −1 123 −1 −6 Vertical shrink Vertical shrink and reflection in the x-axis ϭ 3 2 ϭϪ 2 (c) y 2x (d) y 3x y y 5 6 4 4 3 2 2 x −6 −4 −2 246 1 x −3 −2 −1 123 −1 Vertical stretch Vertical stretch and reflection in the x-axis 10. (a) y ϭ x2 ϩ 1 (b) y ϭ x2 Ϫ 1 y y 5 4 4 3 3 2 2 1 x −3 −2 23 x −3 −2 −1 123 −1 −2 Vertical translation one unit upward Vertical translation one unit downward (c) y ϭ x2 ϩ 3 (d) y ϭ x2 Ϫ 3 y y 10 8 8 6 6 4 x −6–4 46 x −6 −4 −2 246 −2 −4 Vertical translation three units upward Vertical translation three units downward 138 Chapter 2 Polynomial and Rational Functions 11. (a) y ϭ ͑x Ϫ 1͒2 (b) y ϭ ͑3x͒2 ϩ 1 y y 5 5 4 4 3 3 x x − − 2 1 1234 −3 −2 −1 1 23 − 1 −1 Horizontal translation one unit to the right Horizontal shrink and a vertical translation one unit upward ϭ ͑1 ͒2 Ϫ ϭ ͑ ϩ ͒2 (c) y 3x 3 (d) y x 3 y y 8 10 6 8 4 2 x 2 −6 −226 −2 x −8 −6 −4 −224 −4 −2 Horizontal stretch and a vertical translation three units Horizontal translation three units to the left downward ϭϪ1͑ Ϫ ͒2 ϩ ϭ ͓1͑ Ϫ ͔͒ 2 Ϫ 12. (a) y 2 x 2 1 (b) y 2 x 1 3 y y 8 10 6 8 4 6 4 x −6 −4 −22 6810 x −8268−6 −4 −4 −6 Horizontal translation two units to the right, vertical Horizontal translation one unit to the right, horizontal ͑ 1͒ shrink each y-value is multiplied by 2 , reflection in stretch (each x-value is multiplied by 2), and vertical the x-axis, and vertical translation one unit upward translation three units downward ϭϪ1͑ ϩ ͒2 Ϫ 2 (c) y 2 x 2 1 (d) y ϭ ͓2͑x ϩ 1͔͒ ϩ 4 y y 6 7 4 2 4 x −8 −6 −4 264 3 2 −4 1 −6 x −4234−3 −2 −1 1 − −8 1 Horizontal translation two units to the left, vertical Horizontal translation one unit to the left, horizontal ͑ 1͒ ͑ 1͒ shrink each y-value is multiplied by 2 , reflection in shrink each x-value is multiplied by 2 , and vertical trans- x-axis, and vertical translation one unit downward lation four units upward Section 2.1 Quadratic Functions and Models 139 13. f ͑x͒ ϭ x2 Ϫ 5 14. h͑x͒ ϭ 25 Ϫ x2 Vertex: ͑0, Ϫ5͒ Vertex: ͑0, 25͒ Axis of symmetry:x ϭ 0 or the y-axis Axis of symmetry: x ϭ 0 Find x-intercepts: y Find x-intercepts: y 2 x2 Ϫ 5 ϭ 0 Ϫ 2 ϭ 30 1 25 x 0 x 2 ϭ 2 x 5 −4 −3 −11 34 x ϭ 25 ϭ ±Ί −2 x 5 x ϭ ±5 −3 x-intercepts: x-intercepts: ͑±5, 0͒ x −20−10 10 20 ͑ϪΊ5, 0͒, ͑Ί5, 0͒ −6 ͑ ͒ ϭ 1 2 Ϫ ϭ 1͑ Ϫ ͒2 Ϫ ͑ ͒ ϭ ϭ 1 2 ϭϪ1 2 ϩ 15. f x 2x 4 2 x 0 4 16. f x 16 4x 4x 16 Vertex: ͑0, Ϫ4͒ Vertex: ͑0, 16͒ Axis of symmetry:x ϭ 0 or the y-axis Axis of symmetry: x ϭ 0 Find x-intercepts: y Find x-intercepts: y 3 1x2 Ϫ 4 ϭ 0 Ϫ 1 2 ϭ 18 2 2 16 4x 0 2 1 x ϭ 8 2 ϭ 12 x x 64 − − − ϭ ±Ί ϭ ± Ί 4 3 1 1234 9 x 8 2 2 x ϭ ±8 −2 6 x-intercepts: −3 x-intercepts: ͑±8, 0͒ 3 x − − − ͑Ϫ2Ί2, 0͒, ͑2Ί2, 0͒ −5 9 6 3 369 −3 17. f ͑x͒ ϭ ͑x ϩ 5͒2 Ϫ 6 y 18. f ͑x͒ ϭ ͑x Ϫ 6͒2 ϩ 3 y Vertex: ͑Ϫ5, Ϫ6͒ 20 Vertex: ͑6, 3͒ 50 16 40 Axis of symmetry: x ϭϪ5 12 Axis of symmetry: x ϭ 6 30 Find x-intercepts: Find x-intercepts: 20 ͑ ϩ ͒2 Ϫ ϭ x 2 x 5 6 0 −20 −12 4 8 ͑x Ϫ 6͒ ϩ 3 ϭ 0 10 2 2 x ͑x ϩ 5͒ ϭ 6 −8 ͑ Ϫ ͒ ϭϪ x 6 3 −20−10 10 20 30 x ϩ 5 ϭ ±Ί6 Not possible for real x x ϭϪ5 ± Ί6 No x-intercepts x-intercepts: ͑Ϫ5 Ϫ Ί6, 0͒, ͑Ϫ5 ϩ Ί6, 0͒ 19. h͑x͒ ϭ x2 Ϫ 8x ϩ 16 ϭ ͑x Ϫ 4͒2 20. g͑x͒ ϭ x2 ϩ 2x ϩ 1 ϭ ͑x ϩ 1͒2 Vertex: ͑4, 0͒ y Vertex: ͑Ϫ1, 0͒ y Axis of symmetry: x ϭ 4 20 Axis of symmetry: x ϭϪ1 6 5 ͑ ͒ 16 ͑Ϫ ͒ x-intercept: 4, 0 x-intercept: 1, 0 4 12 3 8 2 4 1 x x −4481216 −4 −3 −2 −112 140 Chapter 2 Polynomial and Rational Functions 5 1 21. f͑x͒ ϭ x2 Ϫ x ϩ 22. f ͑x͒ ϭ x2 ϩ 3x ϩ 4 4 1 1 5 9 9 1 ϭ ΂x2 Ϫ x ϩ ΃ Ϫ ϩ ϭ ΂x2 ϩ 3x ϩ ΃ Ϫ ϩ 4 4 4 4 4 4 1 2 3 2 ϭ ΂x Ϫ ΃ ϩ 1 ϭ ΂x ϩ ΃ Ϫ 2 2 2 1 3 Vertex: ΂ , 1΃ Vertex: ΂Ϫ , Ϫ2΃ y 2 2 4 1 3 3 Axis of symmetry: x ϭ y Axis of symmetry: x ϭϪ 2 2 2 5 1 Find x-intercepts: Find x-intercepts: x 4 −5 −4 −3 −2 −112 5 3 1 x2 Ϫ x ϩ ϭ 0 x2 ϩ 3x ϩ ϭ 0 −2 4 4 −3 1 ± Ί1 Ϫ 5 1 Ϫ3 ± Ί9 Ϫ 1 x ϭ x ϭ 2 x 2 −2 −1 123 Not a real number 3 ϭϪ ± Ί2 2 No x-intercepts 3 x-intercepts: ΂Ϫ ± Ί2, 0΃ 2 23. f ͑x͒ ϭϪx2 ϩ 2x ϩ 5 24. f ͑x͒ ϭϪx2 Ϫ 4x ϩ 1 ϭϪ͑x2 ϩ 4x͒ ϩ 1 ϭϪ͑x2 Ϫ 2x ϩ 1͒ Ϫ ͑Ϫ1͒ ϩ 5 ϭϪ͑x2 ϩ 4x ϩ 4͒ Ϫ ͑Ϫ4͒ ϩ 1 ϭϪ͑x Ϫ 1͒2 ϩ 6 ϭϪ͑x ϩ 2͒2 ϩ 5 Vertex: ͑1, 6͒ Vertex: ͑Ϫ2, 5͒ Axis of symmetry: x ϭ 1 Axis of symmetry: x ϭϪ2 Find x-intercepts: Find x-intercepts: Ϫx2 Ϫ 4x ϩ 1 ϭ 0 Ϫx2 ϩ 2x ϩ 5 ϭ 0 x2 ϩ 4x Ϫ 1 ϭ 0 x2 Ϫ 2x Ϫ 5 ϭ 0 Ϫ4 ± Ί16 ϩ 4 x ϭ 2 2 ± Ί4 ϩ 20 x ϭ 2 ϭϪ2 ± Ί5 ϭ 1 ± Ί6 x-intercepts: ͑Ϫ2 ± Ί5, 0͒ x-intercepts: ͑1 Ϫ Ί6, 0͒, ͑1 ϩ Ί6, 0͒ y 5 y 4 6 2 1 x −6 −5 −3 −2 −112 x − −42 6 2 −2 −3 −4 Section 2.1 Quadratic Functions and Models 141 25.