19. f(x)(x- 2)2 + 4 x + 1)3 - 3 3 20. f(x) Use an end behavior diagram, , , , or the graph of each polynomial function. See Example 2. to describe the end behavior of f(x)= 5x5 +2x3-3x + 4 21. 22. f(x) -x3 - 4x2 + 2x- 1 1 23. f(x) = -4x3 + 3x2- 1 24. f(x) = 4x7 - x5 x3 25. f(x) = 9x6- 3x4 +x2 - 2 26. f(x) 10x6 - x5 + 2x - 2 27. f(x) = 3 + 2x- 4x2 - 5x10 f(x) = 7 2x- 5x2- 10x4 Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. 29. f(x)= xr3 +5x2 +2x - 8 30. f(x)= x3 +3x2- 13x - 15 31. f(x) 2x(x - 3) (x + 2) 33. f(x) x2(x- 2) (x + 3)2 32. f(x) x(x + 1)(x - 1) 34. f(x) x2(x- 5) (x + 3) (x - 1) 36. f(x) (4x + 3) (x + 2)2 35. f(x) (3x- 1)(x + 2)2 37. f(x) x3+ 5x2 - x - 5 38. f(x) x3-+x- 36x- 36 39. f(x) x3 - x2 - 2x 1 40. f(x) 3x4 +5x3-2x2 41. f(x) 2x3(x? - 4) (x - 1) 42. f(x) x2(x- 3)'(x + 1) 43. f(x) 2x3 - 5x2 - x + 6 44. f(x) 2x4 + x3 -6x2- 7x - 2 1 45. f(x) = 3x4 7x3-6x2 +12x + 8 46. f(x) = x4 +3x3- 3x2- 11x - 6 Use the intermediate value theorem to show that each polynomial function has a recal between the numbers given. See Example 5. yer 47. f(x) 2x2-7x+ 4; 2 and 3 48. f(x) 3r2-x- 4; l and 49. f (x) = 2x3 - 5x2 -5x+7; 0 and 1 50. f(x) 2x3-9x2+x+ 20; 2 and 2.5 11 51. f(x) 2x4- 4x2 + 4x- 8; 1 and 2 52. f(x) x4-4x3 - x + 3; 0.5 and 1 53. f(x) x4+x3 -6x2-20x - 16; 3.2 and 3.3 54. f(x) x4-2x3-2x2- 18x +5; 3.7 and 3.8 55. f(x) = x4-4r3-20x2+32x + 12; -1 and 0 56. f(x) = x5 + 2xrt +x3+3; -1.8 and-1.7 Show that the real zeros of each polynomial function satisfy the given conditions. See

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.