Intro to factors & divisibility | Mathematics II | High School Math | Khan Academy — Transcript

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1. 0:00You're probably familiar with the
2. 0:01general term factor. So, if I were to
3. 0:04say, "What are the factors of 12?" You
4. 0:08can say, "Well, what are the whole
5. 0:09numbers that I can multiply by another
6. 0:12whole number to get 12?" So, ex- for
7. 0:15examples, you could say things like,
8. 0:17"Well, I could multiply 1 * 12 to get
9. 0:2112." So, you could say that 1 is a
10. 0:23factor of 12. You could even say that 12
11. 0:25is a factor of 12. You could say 2 * 6
12. 0:28is equal to 12. So, you could say that 2
13. 0:30is a factor of 12 and that 6 is also a
14. 0:33factor of 12. And of course, 3 * 4 is
15. 0:36also 12. So, both 3 and 4 are factors of
16. 0:4012. So, if you said, "Well, what are the
17. 0:41factors of 12?" And you've seen this
18. 0:42before, you'd say, "Well, you could say
19. 0:441, 2, 3, 4, 6, and 12." Those are all
20. 0:48factors of 12. And you could also phrase
21. 0:51it the other way around. So, let me just
22. 0:52give an example. So, if I were to pick
23. 0:53on 3, I could say that 3 is a factor
24. 0:59factor of 12. Or, to phrase it slightly
25. 1:03differently, I could say that 12 is
26. 1:05divisible
27. 1:07divisible
28. 1:0912 is divisible by 3.
29. 1:12Now, what I want to do in this video is
30. 1:14extend this idea of being a factor or
31. 1:16divisibility into the algebraic world.
32. 1:20So, for example, if I were to take
33. 1:243
34. 1:26x
35. 1:27y. So, this is a monomial with an
36. 1:30integer coefficient. 3 is an integer
37. 1:33right over here. And if I were to
38. 1:35multiply it with another monomial with
39. 1:37an integer coefficient, I don't know,
40. 1:39let's say * -2
41. 1:42x squared
42. 1:44y to the third power, what is this going
43. 1:47to be equal to?
44. 1:48Well, this would be equal to if we
45. 1:50multiply the coefficients, 3 * -2
46. 1:54is going to be -6.
47. 1:59x * x squared is x to the third power.
48. 2:04And then y * y to the third is y to the
49. 2:08fourth power.
50. 2:09And so what we could say is if we wanted
51. 2:12to say factors of -6 x to the third y to
52. 2:15the fourth, we could say that 3xy is a
53. 2:17factor of this just as an example. So
54. 2:19let me write that down. We could write
55. 2:21that 3 xy is a factor of
56. 2:27is a factor
57. 2:30of
58. 2:326 of -6 x to the third power y to the
59. 2:37fourth or we could phrase that the other
60. 2:39way around. We could say that -6 x to
61. 2:42the third y to the fourth is divisible
62. 2:45by
63. 2:47is
64. 2:48divisible
65. 2:50is divisible
66. 2:52by 3xy. So hopefully you're seeing the
67. 2:55parallels. If I'm taking these two
68. 2:57monomials with integer coefficients and
69. 2:59I multiply them and I get this other, in
70. 3:01this case this other monomial, I could
71. 3:04say that either one of these and there's
72. 3:05actually other factors of this but I
73. 3:07could say either one of these is a
74. 3:09factor of this monomial or we could say
75. 3:11that -6 x to the third y to the fourth
76. 3:13is divisible by one of its factors. And
77. 3:17we could even extend this to binomials
78. 3:20or polynomials. For example, if I were
79. 3:23to take
80. 3:24if I were to take, let me scroll down a
81. 3:27little bit.
82. 3:28Whoops. If I were to take
83. 3:32let me say
84. 3:34x + 3 and I wanted to multiply it times
85. 3:39x
86. 3:40+
87. 3:427 we know that this is going to be equal
88. 3:45to if I were to write it as a trinomial,
89. 3:47it's going to be x * x so x squared and
90. 3:50then it's going to be 3x + 7x, so plus
91. 3:5410x. And if any of this looks familiar,
92. 3:55we have a lot of videos where we go in
93. 3:57detail of multiplying binomials like
94. 3:59this.
95. 4:00And then 3 * 7 is 21.
96. 4:02Plus 21.
97. 4:05And so because I multiplied these two,
98. 4:07in this case, binomials, or we could
99. 4:09consider themselves to be polynomials.
100. 4:11Polynomials are binomials with integer
101. 4:13coefficients. Notice the coefficients
102. 4:15here, they're 1 1. The constants here,
103. 4:18they're all integers. Because I'm
104. 4:20dealing with all integers here, we could
105. 4:22say that either one of these binomials
106. 4:25is a factor of this trinomial. Or we
107. 4:28could say this trinomial is divisible by
108. 4:31either one of these. So let me write
109. 4:32that down. So I could say, I'll just
110. 4:34pick on x + 7. We could say that x + 7
111. 4:38is a factor
112. 4:40is a factor
113. 4:44of
114. 4:46x squared
115. 4:48plus 10x
116. 4:50plus 21. Or we could say that x squared
117. 4:54+ 10x + 21 is divisible by
118. 4:59is
119. 5:01divisible
120. 5:04by I could say x + 3 or I could say x +
121. 5:067 is divisible by
122. 5:09x
123. 5:10plus
124. 5:11seven. And the key is is that both of
125. 5:13these binomials, or even if we were
126. 5:14dealing with polynomials, we are dealing
127. 5:16with things that have integer we're
128. 5:19dealing with things that have integer
129. 5:21coefficients.