Advanced Math Quiz 3.1-3.2 Review Name: Dec. 2014 Use Synthetic Division to divide the first polynomial by the second polynomial. 1. 5x 3 + 6x 2 8 x + 1, x 5 1. Quotient: 2. x 5 10x 3 + 5 x 1, x + 4 2. Quotient: 3. 12x 3 + 5x 2 + 5 x 7, x + 3 4 3. Quotient: 4. 8x 3 4x 2 + 6 x 3, x 1 2 4. Quotient: 5. 2x 5 3x 4 5x 2 10, x 4 5. Quotient: 6. x 6 1, x + 1 6. Quotient:
Use Synthetic Division and the Remainder Theorem to find P(c). 7. P (x) = 2x 3 x 2 + 3x 1, c = 3 7. Remainder with Synthetic Division: Remainder with the Remainder Theorem: 8. P (x) = 6x 3 x 2 + 4x, c = 3 8. Remainder with Synthetic Division: Remainder with the Remainder Theorem: 9. P (x) = x 5 + 20x 2 1, c = 4 9. Remainder with Synthetic Division: Remainder with the Remainder Theorem: 10. P (x) = x 3 + 3x 2 + 5x + 30, c = 8 10. Remainder with Synthetic Division: Remainder with the Remainder Theorem: Use Synthetic Division and the Factor Theorem to determine whether the given binomial is a factor of P(x). 11. P (x) = x 3 + 4x 2 27x 90, x + 6 11. 12. P (x) = 3x 3 + 4x 2 27x 36, x 4 12. 13. P (x) = 16x 4 8x 3 + 9x 2 + 14x + 4, x 1 4 13. 14. P (x) = x 5 + 2x 4 22x 3 50x 2 75x, x 5 14.
Examine the leading term and the degree of the polynomial to determine the far-left and far-right behavior of the graph. 15. f(x) = 2x 4 3x 2 5x + 1 Degree Sign of Leading Coefficient 16. f(x) = 6x 3 9x 2 + 15x 3 Degree Sign of Leading Coefficient 17. f(x) = 1 2 x5 6x 3 12x 2 + 7 Degree Sign of Leading Coefficient 18. f(x) = 3x 4 + 4x 3 5x 2 x + 6 Degree Sign of Leading Coefficient 19. f(x) = 4x + 4 x 2 Degree Sign of Leading Coefficient 20. f(x) = 81 + x 4 Degree Sign of Leading Coefficient
Given the graphs, determine the far-right and far-left behavior. 21. 22. 23. Find the real zeros of each polynomial function by factoring. The number in parentheses to the right of each polynomial indicates the number of real zeros of the given polynomial function. 24. P(x) = x 3 2x 2 24x (3) 24. 25. P(x) = x 4 5x 2 + 4 (4) 25. 26. P(x) = x 4 29x 2 + 100 (4) 26. 27. P(x) = x 3 7x 2 + 10x (3) 27.
Use the Intermediate Value Theorem to verify that P(x) has a zero between a and b. Explain why there is a zero between a and b. 28. P(x) = 4x 3 x 2 6x + 1; a = 0, b = 1 28. 29. P(x) = 5x 3 16x 2 20x + 64; a = 3, b = 3.5 29. 30. P(x) = x 3 x 2; a = 1.5, b = 1.6 30. 31. P(x) = x 3 2x 2 + x 3; a = 2.8, b = 2.7 31. Procedure for graphing: 1. Start by graphing the zeros 2. Then determine whether the graph passes through the zero or hits and bounces off the zero 3. Graph (if possible) the y-intercept 4. Determine the end behavior which way should the arrows go? 5. Create a smooth curve 32. P(x) = (x + 1)(x 2)(x + 5) What is the degree of the polynomial?
33. P(x) = (x 4) 2 (x + 1) What is the degree of the polynomial? 34. P(x) = (x 2) 2 (x + 5) What is the degree of the polynomial?
35. P(x) = x(x 2) 2 What is the degree of the polynomial? 36. P(x) = (x 1) 2 (x + 3) 2 What is the degree of the polynomial?
37. P(x) = (x 2)(x 1) 2 (x + 4) What is the degree of the polynomial? 38. P(x) = x 2 (x + 3) 2 What is the degree of the polynomial?