Math Chapter 3 Test Review Answers

Pre-Calculus Chapter 3 Test Review Evaluate the expression. Write your answers in a + bi form. 1. (3 − 6𝑖) − (6 − 4𝑖) 2. (2 + 𝑖)(3 − 2𝑖) 5. 2−6𝑖 5𝑖 9. (4 + √−16)(8 − √−25) 6. 48−16𝑖 2−2𝑖 Name:___________________________ 3. 𝑖 19 7. 2+√−8 √−2 4. 𝑖 17 8. 10. (√3 − √−4)(√6 − √−8) Sketch a graph of the function using the zeros, end behavior, and multiplicities. (No calculator!) 11. 𝑃(𝑥) = −(𝑥 + 3)2 (𝑥 − 2) 12. 𝑃(𝑥) = (𝑥 − 1)2 (𝑥 + 5) 13. 𝑃(𝑥) = 𝑥 4 + 2𝑥 3 − 8𝑥 2 14. 𝑃(𝑥) = 𝑥 3 − 2𝑥 2 − 9𝑥 + 18 1+√−28 2−√−4 List all possible rational zeros of the polynomial. 15. 𝑃(𝑥) = 𝑥 4 − 4𝑥 3 − 5𝑥 + 8 16. 𝑃(𝑥) = 2𝑥 4 − 9𝑥 3 + 9𝑥 2 + 𝑥 − 12 Use Descartes Rule of Signs to determine the number of possible positive and negative real zeros of the polynomial. 17. 𝑃(𝑥) = 𝑥 5 − 6𝑥 3 − 𝑥 2 + 2𝑥 + 18 18. 𝑃(𝑥) = 𝑥 8 − 𝑥 5 + 𝑥 4 − 𝑥 3 + 𝑥 2 − 𝑥 + 1 In #19-26, find ALL zeros of the function. (Use Graphing Calculator on #21 & 22 to help with synthetic solutions) 19. 𝑃(𝑥) = −𝑥 3 − 4𝑥 2 + 7𝑥 + 10 20. 𝑃(𝑥) = 𝑥 3 + 6𝑥 2 + 7𝑥 − 4 21. 𝑃(𝑥) = 2𝑥 4 + 𝑥 3 + 2𝑥 2 − 3𝑥 − 2 22. 𝑃(𝑥) = 4𝑥 3 − 7𝑥 + 3 In #19-26, find ALL zeros of the function. (Use Graphing Calculator on #21 & 22 to help with synthetic solutions) 23. 𝑔(𝑥) = 2𝑥 4 + 5𝑥 3 − 5𝑥 2 − 5𝑥 + 3 24. ℎ(𝑥) = 2𝑥 4 + 3𝑥 3 − 6𝑥 2 − 6𝑥 + 4 25. 𝑃(𝑥) = 𝑥 4 + 6𝑥 3 + 17𝑥 2 + 28𝑥 + 20 26. 𝑃(𝑥) = 𝑥 5 − 3𝑥 4 − 𝑥 3 + 11𝑥 2 − 12𝑥 + 4 Find the complete factorization of the following polynomials. 27. 𝑃(𝑥) = 𝑥 4 − 5𝑥 2 − 6 28. 𝑃(𝑥) = 4𝑥 3 + 25𝑥 29. 𝑃(𝑥) = 𝑥 3 − 3𝑥 2 − 4𝑥 + 12 30. 𝑃(𝑥) = 𝑥 2 + 81 31. 𝑃(𝑥) = 𝑥 2 − 8𝑥 + 25 Find a polynomial with the given characteristics. 1 2 32. Degree: 3; Zeros: − , 2, and 3; Constant: 12 33. Degree: 2; Zero: 4; Coefficient of 𝑥: −16 34. Degree: 3; Zeros: 4, 3𝑖 35. Degree: 4; Zeros: 2 – 𝑖, 2𝑖; Constant 40 Identify all of the necessary asymptotes, the 𝒙- and 𝒚-intercepts, and the domain and range. Then graph the function. 𝑥 2 −4 36. 𝑃(𝑥) = 𝑥 2 −1 y-intercept: x-intercept(s): HA: SA: VA: Domain: Range: Identify all of the necessary asymptotes, the 𝒙- and 𝒚-intercepts, and the domain and range. Then graph the function. 37. 𝑃(𝑥) = 𝑥 2 +3𝑥−10 𝑥+2 y-intercept: x-intercept(s): HA: SA: VA: Domain: Range: 38. 𝑃(𝑥) = 4𝑥−3 𝑥−1 y-intercept: x-intercept(s): HA: SA: VA: Domain: Range: 39. 𝑃(𝑥) = 2𝑥 2 −9 𝑥−3 y-intercept: x-intercept(s): HA: SA: VA: Domain: Range: Chapter 3 Review Answers Pre-Calculus 1. −3 − 2𝑖 2. 8 − 𝑖 7. 2 − 𝑖√2 8. 11. 3. −𝑖 4. 𝑖 1−2√7 1+2√7 + 8 𝑖 8 12. 13. 1 2 3 2 18. Pos: 6, 4, 2 or 0 neg: 0 19. -1, -5, 2 1 2 2 𝑖 5 6. 16 + 8𝑖 10. −1√2 − 4𝑖√6 17. Pos: 2 or 0 Neg: 3 or 1 20. -4, −1 ± √2 , −3 24. − 2, 1 2 22. x = 1, 2, − 2 23. ± 1, 26. x = 1, 2, -2 27. 𝑃(𝑥) = (𝑥 + √6)(𝑥 − √6)(𝑥 + 𝑖)(𝑥 − 𝑖) 28. 𝑃(𝑥) = 𝑥(2𝑥 + 5𝑖)(2𝑥 − 5𝑖) − 14. 16. ±1, ±2, ±3, ±4, ±6, ±12, ± , ± 3 6 5 9. 52 + 12𝑖 15. ±1, ±2, ±4, ±8 1 5. − , ±√2 1 −1±𝑖√7 2 21. x = 1, − 2, 25. x = -2, −1 ± 2𝑖 29. (𝑥 − 3)(𝑥 + 2)(𝑥 − 2) 30. (𝑥 + 9𝑖)(𝑥 − 9𝑖) 31. (𝑥 − 4 − 3𝑖)(𝑥 − 4 + 3𝑖) 32. 4𝑥 3 − 18𝑥 2 + 14𝑥 + 12 33. 2𝑥 2 − 16𝑥 + 32 34. 𝑃(𝑥) = 𝑥 3 − 4𝑥 2 + 9𝑥 − 36 35. 𝑃(𝑥) = 2𝑥 4 − 8𝑥 3 + 18𝑥 2 − 32𝑥 + 40 36. x-int: (-2, 0)(2, 0); y-int: (0, 4) VA: x = ±1; HA: y = 1; SA: None D: (−∞, −1) ∪ (−1, 1 ) ∪ (1, ∞) R: (−∞, 1) ∪ [4 ,∞) 3 38. x-int: (4 , 0); y-int: (0, 3) VA: x = 1; HA: y = 4; SA: None D: (−∞, 1) ∪ (1, ∞) R: (−∞, 4) ∪ (4, ∞) 37. x-int: (-5, 0)(2, 0); y-int: (0, -5) VA: x = -2; HA: None; SA: y = x + 1 D: (−∞, −2) ∪ (−2, ∞) R: (−∞, ∞) 3√2 39. x-int: (± 2 , 0); y-int: (0, 3) VA: x = 3; HA: None; SA: y = 2x + 6 D: (−∞, 3) ∪ (3, ∞) R: (−∞, 3.5] ∪ [20.5 ,∞)