x 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.
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1 1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x x + 7 have? 3. How many local maxima and minima does the polynomial y = x 4 3 x have? 4. How many local maxima and minima does the following polynomial have? y = 0.2 x x 4 13 x x x Find the x coordinates of all intersection points of the graphs of two functions: y = x 3 4 x 2 8 x + 6 and y = x x An open box is to be constructed from a piece of cardboard a = 20 cm by b = 34 cm by cutting squares of length x from each corner and folding up the sides, as shown in the figure. What is the maximum volume of such a box? 7. Find the quotient and remainder using long division. x 2 + 6x 18 x + 2 PAGE 1
2 8. Find the quotient and remainder using long division 15x x x 3x Find the quotient and remainder using long division 9x 2 6x + 9 3x 2 3x 10. Find the quotient and remainder using long division. x x x x Find the quotient and remainder using synthetic division x 3 + 9x x + 50 x Find the quotient and remainder using synthetic division. 2x 3 + 9x 2 86x 62 x Find the quotient and remainder using synthetic division. x 4 6x x 2 348x 26 x 8 PAGE 2
3 14. Find the quotient and remainder using synthetic division. 2x 4 + 5x 3 + 4x 2 + 3x + 9 x Find the quotient and remainder using synthetic division. x 3 27 x Use synthetic division and the remainder theorem to evaluate P ( 2 ), for P ( x ) = 2 x x x 4 6 x x ( ) 17. Use synthetic division and the remainder theorem to evaluate, for P ( x ) = x 3 6 x x + 4. P Given that x = 5 is zero of P (x) = x 3 3 x 2 25 x + 75, find all other zeros of P (x). 19. Find a polynomial of degree 3 that has zeros 2, 2, Find a polynomial of the specified degree that has the given zeros: degree 5; zeros 4, 1, 0, 1, Find a polynomial of degree 3 that has zeros 1, 3, and 4, and in which the coefficient of x 2 is List all possible rational zeros given by the rational zeros theorem. Don t check to see which actually are zeros. P (x) = x 3 3x PAGE 3
4 23. List all possible rational zeros given by the rational zeros theorem (but don t check to see which actually are zeros). P (x) = 2x 5 + 5x x Find all rational zeros of the polynomial: P(x) = x 3 + 6x Find all rational zeros of the polynomial: P (x) = x 3 3x 2 + 3x Find all rational zeros of the polynomial: P (x) = x 3 + 3x 2 + 9x Find all rational zeros of the polynomial: P (x) = x 4 34x Find all rational zeros of the polynomial: P (x) = 4x 4 13x Find all rational zeros of the polynomial: P (x) = x 4 + 4x 3 + 6x 2 + 4x + 1 PAGE 4
5 30. Find all rational zeros of the polynomial: P(x) = 16x x 2 x Find all the real zeros of the polynomial: Use the quadratic formula if necessary. P (x) = x 3 + 7x 2 + 9x Find all the real zeros of the polynomial: Use the quadratic formula if necessary. P (x) = 24x 3 76x Find all the real zeros of the polynomial: Use the quadratic formula if necessary. P (x) = 3x x x 2 20x Find integers that are upper and lower bounds for the real zeros of the polynomial: P (x) = x 3 24x x Find all rational zeros of the polynomial: P (x) = 2x 4 + 3x 3 52x 2 75x + 50 and then find the irrational zeros, if any. Whenever appropriate, use the rational zeros theorem, the upper and lower bounds theorem, Descartes rule of signs, the quadratic formula, or other factoring techniques. PAGE 5
6 36. Find all rational zeros of the polynomial: P (x) = 25x 4 151x and then find the irrational zeros, if any. Whenever appropriate, use the rational zeros theorem, the upper and lower bounds theorem, Descartes rule of signs, the quadratic formula, or other factoring techniques. 37. Evaluate the expression ( i) + (18 2 i) and write the result in the form a + bi. 38. Evaluate the expression 4( i) and write the result in the form a + bi. 39. Evaluate the expression (4 + 7 i)(8 6 i) and write the result in the form a + bi i 40. Evaluate the expression and write the result in the form a + bi. 4 3i Evaluate the expression and write the result in the form a + bi. 3 6i i 42. Evaluate the expression and write the result in the form a + bi. 4i Evaluate the expression and write the result in the form a + bi. 3 + i 3 i 44. Evaluate the expression i 9 and write the result in the form a + bi. PAGE 6
7 45. Evaluate the expression i 4 and write the result in the form a + bi. 46. Evaluate the expression 4 and write the result in the form a + bi. 47. Evaluate the expression and write the result in the form a + bi. 48. Evaluate the expression (1 + 64)(12 49) and write the result in the form a + bi Evaluate the expression and write the result in the form a + bi Find all solutions of the equation x = 0 and express them in the form a + bi. 51. Find all solutions of the equation x 2 8 x + 25 = 0 and express them in the form a + bi. 52. Find all solutions of the equation 4 x 2 16 x + 32 = 0 and express them in the form a + bi. 53. Find all solutions of the equation z = 0 and express them in the form a + bi. z PAGE 7
8 ANSWER KEY , 4 1,0 1,2 2,2 0,7, x+4, 26 5x+5,4x 15 3,3x x 4 +8x 2 +3,0 11. x 2 +3x+8, x 2 +19x+9, x 3 +2x 2 +44x+4, x 3 +4x 2 +2x+2, x 2 +3x+9, x=3,x= 5 ( x 2) ( x+2) ( x 1) 20. x 5 17x 3 +16x 21. 3x 3 +6x 2 +33x x= 19,x= x= 1,x= 2,x= 38,x= 19,x= 24. x=2,x= x=1 26. x= x= 5,x=5,x= 3,x=3 28. x= 1,x=1,x= 3 2,x= x= x= 2,x= 1 4,x= x= 5,x= 1 2 x= ,x= 6 2 ( ) 33. x= 2,x= 1,x= x= 1,x= x= 2,x= 1,x= 5,x= ,x= x= 1 5,x= i i i i i i i,x= 6,x= 6 PAGE 1
9 ANSWER KEY 44. i i i 49. i i, 8i i,4 3i i,2 2i i, 1 4i PAGE 2
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