Ex 1: Find the Zeros of a Polynomial Function - Integer Zeros — Transcript

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1. 0:00WELCOME TO THE FIRST OF SEVERAL EXAMPLES
2. 0:02OF FINDING THE ZEROS OF A POLYNOMIAL FUNCTION.
3. 0:05THERE ARE TWO MAIN IDEAS WE NEED TO BE AWARE OF
4. 0:07TO HELP US FIND THE ZEROS OR ROOTS
5. 0:09OF A POLYNOMIAL FUNCTION.
6. 0:11FIRST IF WE'RE ALLOWED TO GRAPH THE POLYNOMIAL FUNCTION,
7. 0:15THEN THE REAL ZEROS OF THE FUNCTION
8. 0:17WILL BE THE X INTERCEPTS OF THE GRAPH.
9. 0:20SO WHILE EVEN THOUGH ALL OF THE REAL 0'S
10. 0:22WILL SHOW AS X INTERCEPTS,
11. 0:23IF THEY'RE REAL AND RATIONAL
12. 0:25WE CAN USE THESE TO HELP US FIND THE 0'S
13. 0:27AND ALSO TO WRITE THE FUNCTION IN FACTORED FORM.
14. 0:30IF WE'RE NOT ALLOWED TO USE THE GRAPH
15. 0:32TO HELP US FIND THE 0'S OF THE FUNCTION,
16. 0:34THEN WE'LL HAVE TO USE THE RATIONAL 0 --
17. 0:37WHICH SAYS IF A POLYNOMIAL FUNCTION
18. 0:39HAS REAL RATIONAL 0'S,
19. 0:42THEY WILL BE A RATIO OF THE FACTORS OF THE CONSTANT TERM
20. 0:45TO THE FACTORS OF THE LEADING COEFFICIENT.
21. 0:48SO WHAT THAT MEANS IS IF WE LOOK AT THIS EXAMPLE HERE,
22. 0:51IF THIS FUNCTION DOES HAVE REAL RATIONAL 0'S,
23. 0:54IT WOULD HAVE TO BE A RATIO OF THE FACTORS OF 6
24. 0:58TO THE FACTORS OF +2.
25. 1:00WELL THE FACTORS OF 6 WITH THE NUMBERS
26. 1:02THAT DIVIDE EVENLY INTO 6
27. 1:04WOULD BE + OR - 1, + OR - 2, + OR - 3 AND + OR - 6.
28. 1:14AND THE FACTORS OF 2 WOULD BE + OR - 1 AND + OR - 2.
29. 1:20SO THE RATIO OF THESE FACTORS
30. 1:23WOULD GIVE US POSSIBLE RATIONAL 0'S
31. 1:25OF THE GIVEN POLYNOMIAL.
32. 1:27SO LET'S GO AHEAD AND LIST THESE.
33. 1:29IF WE HAVE A DENOMINATOR OF + OR - 1,
34. 1:31IT'S NOT GOING TO AFFECT THE FACTORS OF THE NUMERATOR
35. 1:34SO WE'D HAVE FACTORS OF + OR - 1, + OR - 2,
36. 1:39+ OR - 3 AND + OR - 6.
37. 1:43BUT NOW IF WE USE A DENOMINATOR OF 2
38. 1:46FROM THESE FACTORS HERE,
39. 1:48NOTICE HOW WE'D HAVE + OR - 1/2,
40. 1:50THAT'S + OR - ONE HALF, + OR - 2/2 THAT'S + OR - 1
41. 1:57WHICH WE ALREADY HAVE.
42. 1:58+ OR - 3 HALVES AND THEN WE'D HAVE + OR - 6/2
43. 2:06WHICH IS + OR - 3 WHICH WE ALREADY HAVE.
44. 2:09SO THE RATIONAL 0'S OF THIS FUNCTION
45. 2:12MUST COME FROM THIS LIST.
46. 2:14SO FROM HERE, WE'D HAVE TO USE TRIAL AND ERROR
47. 2:17TO SEE WHICH OF THESE WOULD MAKE THE FUNCTION EQUAL TO 0.
48. 2:20SO IF WE ARE ALLOWED TO USE THE GRAPH,
49. 2:22IT'S EXTREMELY HELPFUL.
50. 2:25FOR THIS FIRST BASIC EXAMPLE, WE'LL LOOK AT IT BOTH WAYS.
51. 2:28WE'LL FIND THE 0'S USING THE GRAPH
52. 2:30AND THEN WE'LL ALSO FIND THE ZEROS
53. 2:31USING THE RATIONAL ROOT THEOREM
54. 2:33JUST SO YOU CAN SEE THE DIFFERENCE.
55. 2:35SO IF WE GRAPH THIS FUNCTION,
56. 2:38WE'RE LOOKING FOR THE X INTERCEPTS.
57. 2:40NOTICE HOW THIS FUNCTION
58. 2:41AND CROSSES THE X AXIS AT -3 +1 AND +5
59. 2:48WHICH ARE THE 0'S OF THIS FUNCTION.
60. 2:52SO YOU CAN SEE THIS GRAPH IS EXTREMELY HELPFUL
61. 2:53IN THIS CASE.
62. 2:54IT GAVE US ALL OF THE 0'S.
63. 2:59NOW WE SHOULD MAKE THE CONNECTION
64. 3:01THAT IF THESE ARE THE 0'S,
65. 3:02THIS WOULD HELP US WRITE THIS FUNCTION IN FACTORED FORM
66. 3:05OR FUNCTION F OF X WOULD HAVE TO HAVE A FACTOR OF X + 3
67. 3:10FROM THE 0 OF -3.
68. 3:13WE'D HAVE TO HAVE A FACTOR OF X - 1 FROM THE 0 OF +1
69. 3:18AND WE'D ALSO HAVE TO HAVE A FACTOR OF X - 5
70. 3:20FROM THE 0 OF +5.
71. 3:23AGAIN GRAPHICALLY WE FOUND THE 0'S VERY EASILY
72. 3:25BECAUSE THEY WERE ALL REAL AND RATIONAL.
73. 3:29NOW ASSUMING WE COULD NOT LOOK AT THE GRAPH,
74. 3:31WE WOULD HAVE TO USE THE RATIONAL ROOT THEOREM
75. 3:33TO FIND THESE 0'S.
76. 3:35SO WE'D HAVE TO FIRST START WITH A LIST
77. 3:36OF POSSIBLE RATIONAL 0'S
78. 3:38WHICH AGAIN WOULD BE THE FACTORS OF THE CONSTANT TERM
79. 3:41IN THIS CASE 15 OVER THE FACTORS
80. 3:43OF THE LEADING COEFFICIENT
81. 3:45WHICH IN THIS CASE IS JUST 1.
82. 3:47SO THE FACTORS OF 15 WOULD BE + OR - 1, + OR - 3, + OR - 5,
83. 3:55AND + OR - 15.
84. 3:57OF COURSE THE FACTORS OF 1 JUST + OR - 1
85. 4:01WE FORM A RATIO HERE.
86. 4:04HAVING A DENOMINATOR OF + OR - 1
87. 4:05IS NOT GOING TO AFFECT OUR POSSIBLE RATIONAL 0'S
88. 4:08FROM THE FACTORS OF 15,
89. 4:11SO THE POSSIBLE RATIONAL O'S WOULD HAVE TO BE + OR - 1,
90. 4:16+ OR - 3, + OR - 5 AND + OR - 15.
91. 4:21SO FROM HERE WE HAVE TWO WAYS
92. 4:23TO DETERMINE WHICH OF THESE WOULD BE THE RATIONAL 0'S.
93. 4:26WE CAN -- THEM INTO THE FUNCTION
94. 4:28AND SEE WHICH MAKE THE FUNCTION EQUAL TO 0
95. 4:31OR WE CAN USE THESE VALUES TO PERFORM SYNTHETIC DIVISION
96. 4:35AND SEE WHICH WOULD GIVE A REMAINDER OF 0.
97. 4:38SO LET'S USE SYNTHETIC DIVISION
98. 4:39AND LET'S START WITH A POSSIBLE 0 OF LET'S SAY -1.
99. 4:43SO WE'D PERFORM SYNTHETIC DIVISION,
100. 4:44WE'D HAVE -1 HERE IN THIS LITTLE BOX
101. 4:47AND THEN WE LIST THE COEFFICIENTS OF THE POLYNOMIAL
102. 4:49SO WE HAVE 1, -3, -13 AND +15.
103. 4:57AND NOW WE'RE HOPING THE REMAINDER IS 0.
104. 5:00AGAIN THIS IS ASSUMING
105. 5:01WE DIDN'T ALREADY FIND THE 0'S GRAPHICALLY.
106. 5:03SO I BRING THE 1 DOWN, MULTIPLY BY -1, THAT'S -1,
107. 5:08ADD, MULTIPLY BY -1 AGAIN, THAT'S +4,
108. 5:13ADD, MULTIPLY BY -1 AGAIN THAT'S + 9, ADD, THAT'S 24.
109. 5:21WE HAVE A REMAINDER OF 24 HERE
110. 5:23WHICH MEANS -1 IS NOT A 0 OF THE POLYNOMIAL FUNCTION.
111. 5:27SO LET'S TRY A +1.
112. 5:29BRING THE 1 DOWN AGAIN, MULTIPLY IT BY A +1 THIS TIME
113. 5:32THAT'S 1
114. 5:33ADD, MULTIPLY BY +1 THAT'S -2,
115. 5:38ADD, MULTIPLY BY +1 AND ADD.
116. 5:43AND THIS IS GOOD NEWS BECAUSE OUR REMAINDER IS 0,
117. 5:47+1 IS A 0 OF THE FUNCTION.
118. 5:55BUT BECAUSE IT'S A CUBIC WE'RE STILL LOOKING FOR TWO MORE 0'S
119. 5:58BUT WE CAN USE THIS 0 TO HELP US WRITE THE POLYNOMIAL
120. 6:01IN FACTORED FORM.
121. 6:03WE NOW KNOW THAT F OF X IS EQUAL TO,
122. 6:08WELL IF 1 IS A 0, X - 1 MUST BE A FACTOR
123. 6:12AND BECAUSE WE PERFORM SYNTHETIC DIVISION,
124. 6:14WE KNOW THAT THE OTHER FACTOR
125. 6:16WOULD HAVE TO BE THIS DEGREE 2 POLYNOMIAL
126. 6:18WHICH WOULD BE 1X SQUARED - 2X - 15.
127. 6:27THIS IS THE REASON WHY PERFORMING SYNTHETIC DIVISION
128. 6:29IS A NICE WAY TO DETERMINE THE 0'S
129. 6:32BECAUSE IF THE REMAINDER IS 0,
130. 6:34IT GIVES US TWO FACTORS OF THE POLYNOMIAL FUNCTION
131. 6:37AND NOW WE CAN TRY TO FACTOR THIS
132. 6:39AND IF IT DOESN'T FACTOR,
133. 6:41WE'D HAVE TO USE THE QUADRATIC FORMULA.
134. 6:43AGAIN WE'RE TRYING TO FIND THE X VALUES
135. 6:44THAT MAKE THIS EQUAL TO 0.
136. 6:47SO WE'D HAVE X - 1.
137. 6:49THIS DOES FACTOR, THE FACTORS OF -15 AND ADD THE -2
138. 6:57THAT WOULD BE -5 AND +3.
139. 7:02SO IF WE'RE LOOKING FOR THE VALUE OF X
140. 7:03THAT MAKE THIS EQUAL TO 0,
141. 7:06AGAIN WE ALREADY FOUND X = 1 IS A 0.
142. 7:09HERE WE HAVE X = 5 AS A 0 AND HERE WE HAVE X = -3 AS A 0.
143. 7:16NOTICE HOW THESE 3 X VALUES WOULD BE THE 0'S
144. 7:19THAT WE'RE LOOKING FOR
145. 7:20SO WE HAVE +1, +5 AND -3 WHICH ARE THE EXACT SAME 0'S
146. 7:26WE FOUND GRAPHICALLY VERY EASILY.
147. 7:31SO I'M SURE YOU AGREE IF YOU HAD A CHOICE
148. 7:33BEING ABLE TO GRAPH THE FUNCTION IS EXTREMELY HELPFUL,
149. 7:35IT CAN SAVE QUITE A BIT OF TIME,
150. 7:37BUT IF WE'RE NOT ABLE TO GRAPH THE FUNCTION,
151. 7:39WE WOULD HAVE TO USE THIS METHOD HERE.
152. 7:41WE'LL TAKE A LOOK AT SEVERAL MORE EXAMPLES
153. 7:43IN THE NEXT FEW VIDEOS MANY OF THEM WE'LL USE THE COMBINATION
154. 7:46OF USING THE GRAPH AS WELL AS THE METHOD WE USED HERE.
155. 7:49I HOPE YOU FOUND THIS HELPFUL.