Ex 1: Divide a Trinomial by a Binomial Using Long Division — Transcript

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1. 0:01- NOW WE'RE GOING TO LOOK AT 2 EXAMPLES
2. 0:02OF DIVIDING A POLYNOMIAL BY A BINOMIAL, OR MORE SPECIFICALLY,
3. 0:06A DEGREE 2 TRINOMIAL DIVIDED BY A DEGREE 1 BINOMIAL.
4. 0:11AND WE'RE GOING TO USE THE METHOD OF LONG DIVISION,
5. 0:13AND SINCE A FRACTION BAR REPRESENTS DIVISION,
6. 0:16WE CAN WRITE THIS AS X SQUARED - 5X + 7 DIVIDED BY X - 2.
7. 0:30X SQUARED - 5X + 7 IS CALLED THE DIVIDEND,
8. 0:33AND THESE TERMS MUST BE IN DESCENDING ORDER,
9. 0:36MEANING FROM HIGHEST DEGREE TO LOWEST DEGREE, WHICH THEY ARE.
10. 0:39AND THE SAME THING FOR X - 2, WHICH IS THE DIVISOR.
11. 0:43AND OUR ANSWER WILL BE THE QUOTIENT.
12. 0:46NOW THERE'S A COUPLE OF WAYS TO START THIS DIVISION PROCESS,
13. 0:48BUT WE ARE ONLY CONCERNED ABOUT THE FIRST TERM OF THE DIVIDEND
14. 0:53AND THE FIRST TERM OF THE DIVISOR.
15. 0:55ONE WAY TO DETERMINE THE FIRST TERM OF THE QUOTIENT
16. 0:56IS TO ASK WHAT x X = X SQUARED?
17. 1:01SO IF WE KNOW X x X = X SQUARED,
18. 1:04THE FIRST TERM OF OUR QUOTIENT WILL BE X.
19. 1:09HOWEVER, THIS IS A DIVISION PROBLEM,
20. 1:11SO ANOTHER WAY TO DETERMINE THIS FIRST TERM
21. 1:13WOULD BE TO TAKE X SQUARED AND DIVIDE BY X.
22. 1:17AND NOTICE THAT X SQUARED DIVIDED BY X WOULD ALSO EQUAL X,
23. 1:22OUR FIRST TERM.
24. 1:24NOW, ONCE WE HAVE THIS X HERE
25. 1:25WE'RE GOING TO MULTIPLY IT WITH THE DIVISOR,
26. 1:28AND THEN PUT IT BELOW.
27. 1:32SO WE HAVE X x X = X SQUARED, AND THEN X x -2 = -2X.
28. 1:39SO WE HAVE -2X.
29. 1:42AND THEN WHEN PERFORMING LONG DIVISION
30. 1:44WE NEED TO SUBTRACT THIS RESULT,
31. 1:48BUT INSTEAD OF SUBTRACTING THIS BINOMIAL,
32. 1:50IT'S OFTEN MUCH EASIER TO JUST ADD THE OPPOSITE.
33. 1:53SO WE CAN CHANGE THIS TO ADDITION
34. 1:55AS LONG AS WE CHANGE THE SIGN OF EACH OF THE TERMS
35. 1:58INSIDE THE PARENTHESES.
36. 1:59SO WE CAN CHANGE THIS TO ADDITION
37. 2:02AS LONG AS WE CHANGE THIS TO A - X SQUARED,
38. 2:05AND THIS WOULD BECOME 2X.
39. 2:08NOTICE HOW HERE WE HAVE X SQUARED + -X SQUARED.
40. 2:11THAT'S 0.
41. 2:14AND WHEN WE DO THIS, THE FIRST TERM SHOULD ALWAYS BE OPPOSITE.
42. 2:16SO WE HAVE A SUM OF 0, AND THEN WE HAVE -5X + 2X.
43. 2:20THAT'S GOING TO BE -3X.
44. 2:22AND THEN WE BRING DOWN THE NEXT TERM, + 7.
45. 2:27AND NOW WE'RE ONLY CONCERNED
46. 2:28ABOUT THE FIRST TERM OF THE DIVISOR
47. 2:30AND THE FIRST TERM OF THIS RESULT.
48. 2:32AND AGAIN, THERE'S A COUPLE OF WAYS TO FIND THE NEXT TERM.
49. 2:35WE CAN ASK OURSELVES WHAT x X = -3X?
50. 2:40IF WE KNOW THAT WOULD BE -3,
51. 2:42WE CAN GO AHEAD AND PUT -3 UP HERE,
52. 2:44OR SINCE THIS IS A DIVISION PROBLEM,
53. 2:48WE COULD TAKE THE TERM -3X AND DIVIDE BY X.
54. 2:53AND NOTICE HOW THAT ALSO GIVES US THE NEXT TERM OF -3.
55. 2:58AND NOW WE'RE GOING TO MULTIPLY -3 IN THE DIVISOR.
56. 3:01SO WE'LL HAVE -3 x X.
57. 3:04THAT'S -3X.
58. 3:07-3 x -2 = 6, SO WE HAVE 6.
59. 3:12AND AGAIN, WE NEED TO SUBTRACT THIS,
60. 3:16BUT INSTEAD OF SUBTRACTING THIS WE'RE GOING TO ADD THE OPPOSITE.
61. 3:19SO WE'LL CHANGE THIS SIGN TO ADDITION,
62. 3:21CHANGE THIS NEGATIVE TO A POSITIVE,
63. 3:24AND CHANGE THIS 6 TO A NEGATIVE.
64. 3:27AGAIN, NOTICE HOW THE FIRST TERMS
65. 3:28WE HAVE THE SUM OF OPPOSITES.
66. 3:30SO THIS WOULD BE 0, AND THEN WE HAVE 7 + -6.
67. 3:33THAT'S 1.
68. 3:36WE'VE USED ALL THE TERMS, WHICH MEANS THIS IS THE REMAINDER.
69. 3:40WHEN PERFORMING LONG DIVISION WITH POLYNOMIALS,
70. 3:43SINCE THIS REMAINDER IS POSITIVE,
71. 3:44WE'RE GOING TO HAVE PLUS THE REMAINDER/DIVISOR OF X - 2.
72. 3:51SO THIS QUOTIENT IS EQUAL TO X - 3 + 1/X - 2.
73. 3:57ONE MORE THING TO MENTION IN THIS EXAMPLE,
74. 3:59SINCE THERE IS A REMAINDER,
75. 4:01X - 2 DOES NOT DIVIDE EVENLY INTO X SQUARED - 5X + 7,
76. 4:07AND THEREFORE X - 2 IS NOT A FACTOR OF THIS TRINOMIAL.
77. 4:12LOOKING AT OUR SECOND EXAMPLE,
78. 4:13AGAIN, SINCE A FRACTION BAR REPRESENTS DIVISION,
79. 4:16WE CAN WRITE THIS AS X SQUARED + 8X + 12 DIVIDED BY X + 2.
80. 4:26AND AGAIN, TO START, WE'RE ONLY CONCERNED ABOUT THE FIRST TERM
81. 4:29OF THE DIVISOR AND THE FIRST TERM OF THE DIVIDEND.
82. 4:33SO AGAIN, THERE'S TWO WAYS TO DETERMINE THE FIRST TERM
83. 4:35OF OUR QUOTIENT.
84. 4:37WE CAN ASK WHAT x X = X SQUARED?
85. 4:41IF WE KNOW THAT WOULD BE X, WE CAN PUT AN X UP HERE.
86. 4:44THE OTHER WAY IS TO TAKE X SQUARED AND DIVIDE IT BY X,
87. 4:50WHICH IS ALSO EQUAL TO X, GIVING US OUR FIRST TERM.
88. 4:53NOW WE'RE GOING TO MULTIPLY X AND OUR DIVISOR.
89. 4:56SO WE HAVE X x X.
90. 4:57THAT WOULD BE X SQUARED.
91. 5:01X x 2, THAT'S + 2X.
92. 5:05AND WE'RE SUPPOSED TO SUBTRACT THIS,
93. 5:08BUT INSTEAD OF SUBTRACTING THIS
94. 5:09WE'RE GOING TO ADD THE OPPOSITES.
95. 5:11SO WE'LL CHANGE THIS SIGN HERE TO PLUS CHANGE THIS TO NEGATIVE
96. 5:15AND CHANGE THIS TO A NEGATIVE AS WELL.
97. 5:17X SQUARED + -X SQUARED = 0, AND THEN 8 + -2X = 6X.
98. 5:24BRING DOWN THE NEXT TERM, + 12.
99. 5:28AND NOW WE'RE ONLY CONCERNED ABOUT THE X OF THE DIVISOR
100. 5:31AND THE 6X OF THIS RESULT.
101. 5:35SO WE WANT TO KNOW X x WHAT WOULD BE EQUAL TO 6X.
102. 5:38THAT WOULD BE 6, SO WE HAVE + 6,
103. 5:43OR THE ALTERNATIVE WOULD BE TO TAKE 6X AND DIVIDE BY X,
104. 5:48WHICH WOULD ALSO GIVE US 6.
105. 5:51NOW WE'RE GOING TO MULTIPLY 6 IN THE DIVISOR,
106. 5:53SO THAT WOULD BE 6X + 12.
107. 6:00AND WE NEED TO SUBTRACT THIS,
108. 6:03BUT INSTEAD WE'RE GOING TO ADD THE OPPOSITE.
109. 6:05SO WE'LL CHANGE THIS TO POSITIVE, CHANGE THE SIGN HERE,
110. 6:07AND CHANGE THE SIGN HERE.
111. 6:10SO WE HAVE 6X + -6X.
112. 6:12THAT'S 0.
113. 6:13AND WE ALSO HAVE 12 + -12.
114. 6:15THAT'S ALSO 0.
115. 6:17SO HERE THE REMAINDER = 0,
116. 6:21SO OUR QUOTIENT IS X + 6.
117. 6:25BECAUSE THE REMAINDER IS 0
118. 6:27WE CAN SAY X + 2 DIVIDES EVENLY INTO X SQUARED + 8X + 12,
119. 6:32AND THEREFORE X + 2 IS A FACTOR OF OUR TRINOMIAL.
120. 6:38OKAY, WE'LL BE LOOKING AT SEVERAL MORE EXAMPLES
121. 6:39OF LONG DIVISION IN THE NEXT FEW VIDEOS.
122. 6:42I HOPE YOU FOUND THIS HELPFUL.