More examples of factoring by grouping | Algebra I | Khan Academy — Transcript

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1. 0:00In this video, I want to focus on a few more techniques for
2. 0:03factoring polynomials.
3. 0:05And in particular, I want to focus on quadratics that don't
4. 0:09have a 1 as the leading coefficient.
5. 0:11For example, if I wanted to factor 4x squared
6. 0:15plus 25x minus 21.
7. 0:20Everything we've factored so far, or all of the quadratics
8. 0:23we've factored so far, had either a 1 or negative 1 where
9. 0:27this 4 is sitting.
10. 0:28All of a sudden now, we have this 4 here.
11. 0:30So what I'm going to teach you is a technique called,
12. 0:32factoring by grouping.
13. 0:35And it's a little bit more involved than what we've
14. 0:37learned before, but it's a neat trick.
15. 0:40To some degree, it'll become obsolete once you learn the
16. 0:42quadratic formula, because, frankly, the quadratic formula
17. 0:44is a lot easier.
18. 0:46But this is how it goes.
19. 0:47I'll show you the technique.
20. 0:48And then at the end of this video, I'll actually show you
21. 0:50why it works.
22. 0:52So what we need to do here, is we need to think of two
23. 0:55numbers, a and b, where a times b is equal 4 times
24. 1:02negative 21.
25. 1:03So a times b is going to be equal to 4 times negative 21,
26. 1:12which is equal to negative 84.
27. 1:15And those same two numbers, a plus b, need
28. 1:20to be equal to 25.
29. 1:24Let me be very clear.
30. 1:25This is the 25, so they need to be equal to 25.
31. 1:29This is where the 4 is.
32. 1:30So we go, 4 times negative 21.
33. 1:35That's a negative 21.
34. 1:37So what two numbers are there that would do this?
35. 1:40Well, we have to look at the factors of negative 84.
36. 1:45And once again, one of these are going
37. 1:46to have to be positive.
38. 1:47The other ones are going to have to be negative, because
39. 1:50their product is negative.
40. 1:52So let's think about the different
41. 1:53factors that might work.
42. 1:554 and negative 21 look tantalizing, but when you add
43. 1:59them, you get negative 17.
44. 2:01Or, if you had negative 4 and 21, you'd get positive 17.
45. 2:04Doesn't work.
46. 2:06Let's try some other combinations.
47. 2:081 and 84, too far apart when you take their difference.
48. 2:12Because that's essentially what you're going to do, if
49. 2:13one is negative and one is positive.
50. 2:15Too far apart.
51. 2:16Let's see you could do 3-- I'm jumping the gun.
52. 2:202 and 42.
53. 2:23Once again, too far apart.
54. 2:24Negative 2 plus 42 is 40.
55. 2:262 plus negative 42 is negative 40-- too far apart.
56. 2:293 and-- Let's see, 3 goes into 84-- 3 goes into 8 2 times.
57. 2:412 times 3 is 6.
58. 2:428 minus 6 is 2.
59. 2:45Bring down the 4.
60. 2:46Goes exactly 8 times.
61. 2:49So 3 and 28.
62. 2:51This seems interesting.
63. 2:57And remember, one of these has to be negative.
64. 2:59So if we have negative 3 plus 28, that is equal to 25.
65. 3:06Now, we've found our two numbers.
66. 3:08But it's not going to be quite as simple of an operation as
67. 3:12what we did when this was a 1 or negative 1.
68. 3:19What we're going to do now is split up this term right here.
69. 3:29We're going to split it up into positive 28x minus 3x.
70. 3:35We're just going to split that term.
71. 3:37That term is that term right there.
72. 3:39And of course, you have your minus 21 there, and you have
73. 3:43your 4x squared over here.
74. 3:46Now, you might say, how did you pick the 28 to go here,
75. 3:49and the negative 3 to go there?
76. 3:50And it actually does matter.
77. 3:52The way I thought about it is 3 or negative 3, and 21 or
78. 3:56negative 21 , they have some common factors.
79. 3:59In particular, they have the factor 3 in common.
80. 4:02And 28 and 4 have some common factors.
81. 4:04So I grouped the 28 on the side of the 4.
82. 4:07And you're going to see what I mean in a second.
83. 4:09If we, literally, group these so that term becomes 4x
84. 4:15squared plus 28x.
85. 4:18And then, this side, over here in pink, it's plus
86. 4:25negative 3x minus 21.
87. 4:28Once again, I picked these.
88. 4:30I grouped the negative 3 with the 21, or the negative 21,
89. 4:34because they're both divisible by 3.
90. 4:36And I grouped the 28 with the 4, because they're both
91. 4:38divisible by 4.
92. 4:40And now, in each of these groups, we factor as
93. 4:43much out as we can.
94. 4:45So both of these terms are divisible by 4x.
95. 4:49So this orange term is equal to 4x times x-- 4x squared
96. 4:55divided by 4x is just x-- plus 28x divided by 4x is just 7.
97. 5:03Now, this second term.
98. 5:04Remember, you factor out everything that
99. 5:05you can factor out.
100. 5:07Well, both of these terms are divisible by 3 or negative 3.
101. 5:10So let's factor out a negative 3.
102. 5:11And this becomes x plus 7.
103. 5:16And now, something might pop out at you.
104. 5:18We have x plus 7 times 4x plus, x plus 7
105. 5:25times negative 3.
106. 5:27So we can factor out an x plus 7.
107. 5:31This might not be completely obvious.
108. 5:33You're probably not used to factoring
109. 5:34out an entire binomial.
110. 5:36But you could view this could be like a.
111. 5:38Or if you have 4xa minus 3a, you would be able to
112. 5:42factor out an a.
113. 5:43And I can just leave this as a minus sign.
114. 5:50Let me delete this plus right here.
115. 5:51Because it's just minus 3, right?
116. 5:55Plus negative 3, same thing as minus 3.
117. 5:58So what can we do here?
118. 6:00We have an x plus 7, times 4x.
119. 6:02We have an x plus 7, times negative 3.
120. 6:04Let's factor out the x plus 7.
121. 6:06We get x plus 7, times 4x minus 3.
122. 6:16Minus that 3 right there.
123. 6:18And we've factored our binomial.
124. 6:22Sorry, we've factored our quadratic by grouping.
125. 6:25And we factored it into two binomials.
126. 6:28Let's do another example of that, because it's a little
127. 6:30bit involved.
128. 6:30But once you get the hang of it's kind of fun.
129. 6:34So let's say we want to factor 6x squared plus 7x plus 1.
130. 6:43Same drill.
131. 6:44We want to find a times b that is equal to 1 times 6, which
132. 6:50is equal to 6.
133. 6:52And we want to find an a plus b needs to be equal to 7.
134. 6:57This is a little bit more straightforward.
135. 6:59What are the-- well, the obvious one is 1 and 6, right?
136. 7:031 times 6 is 6.
137. 7:051 plus 6 is 7.
138. 7:07So we have a is equal to 1.
139. 7:10Or let me not even assign them.
140. 7:11The numbers here are 1 and 6.
141. 7:15Now, we want to split this into a 1x and a 6x.
142. 7:19But we want to group it so it's on the side of something
143. 7:22that it shares a factor with.
144. 7:24So we're going to have a 6x squar ed here, plus-- and so
145. 7:28I'm going to put the 6x first because 6
146. 7:32and 6 share a factor.
147. 7:34And then, we're going to have plus 1x, right?
148. 7:376x plus 1x equals 7x .
149. 7:39That was the whole point.
150. 7:40They had to add up to 7 .
151. 7:42And then we have the final plus 1 there.
152. 7:46Now, in each of these groups, we can factor out
153. 7:49as much as we like.
154. 7:50So in this first group, let's factor out a 6x.
155. 7:53So this first group becomes 6x times-- 6x squar ed divided by
156. 7:586x is just an x.
157. 7:596x divided by 6x is just a 1.
158. 8:04And then, the second group-- we're going
159. 8:06to have a plus here.
160. 8:08But this second group, we just literally have a x plus 1.
161. 8:11Or we could even write a 1 times an x plus 1.
162. 8:16You could imagine I just factored out of 1 so to speak.
163. 8:19Now, I have 6x times x plus 1, plus 1 times x plus 1.
164. 8:24Well, I can factor out the x plus 1.
165. 8:27If I factor out an x plus 1, that's equal to x plus 1 times
166. 8:356x plus that 1.
167. 8:37I'm just doing the
168. 8:38distributive property in reverse.
169. 8:41So hopefully you didn't find that too bad.
170. 8:43And now, I'm going to actually explain why this little
171. 8:45magical system actually works.
172. 8:50Let me take an example.
173. 8:52I'll do it in very general terms.
174. 8:54Let's say I had ax plus b, times cx-- actually, I'm
175. 9:03afraid to use the a's and b's.
176. 9:05I think that'll confuse you, because I
177. 9:07use a's and b's here.
178. 9:08They won't be the same thing.
179. 9:10So let me use completely different letters.
180. 9:13Let's say I have fx plus g, times hx plus, I'll use j
181. 9:24instead of i.
182. 9:25You'll learn in the future why don't like
183. 9:26using i as a variable.
184. 9:28So what is this going to be equal to?
185. 9:31Well, it's going to be fx times hx which is fhx.
186. 9:35And then, fx times j.
187. 9:37So plus fjx.
188. 9:41And then, we're going to have g times hx.
189. 9:45So plus ghx.
190. 9:49And then g times j.
191. 9:50Plus gj.
192. 9:53Or, if we add these two middle terms, you have fh times x,
193. 10:02plus-- add these two terms-- fj plus gh x.
194. 10:08Plus gj.
195. 10:12Now, what did I do here?
196. 10:14Well, remember, in all of these problems where you have
197. 10:17a non-1 or non-negative 1 coefficient here, we look for
198. 10:20two numbers that add up to this, whose product is equal
199. 10:23to the product of that times that.
200. 10:25Well, here we have two numbers that add up-- let's say that a
201. 10:31is equal to fj.
202. 10:36That is a.
203. 10:37And b is equal to gh.
204. 10:40So a plus b is going to be equal to that middle
205. 10:43coefficient.
206. 10:48And then what is a times b? a times b is going to be equal
207. 10:52to fj times gh.
208. 10:59We could just reorder these terms. We're just multiplying
209. 11:01a bunch of terms. So that could be rewritten as f times
210. 11:04h times g times j.
211. 11:09These are all the same things.
212. 11:11Well, what is fh times gj?
213. 11:14This is equal to fh times gj.
214. 11:19Well, this is equal to the first coefficient times the
215. 11:22constant term.
216. 11:23So a plus b will be equal to the middle coefficient.
217. 11:27And a times b will equal the first coefficient times the
218. 11:31constant term.
219. 11:32So that's why this whole factoring by grouping even
220. 11:37works, or how we're able to figure out what
221. 11:40a and b even are.
222. 11:42Now, I'm going to close up with something slightly
223. 11:44different, but just to make sure that you have a
224. 11:45well-rounded education in factoring things.
225. 11:49What I want to do is to teach you to factor things a little
226. 11:52bit more completely.
227. 11:52And this is a little bit of a add-on.
228. 11:55I was going to make a whole video on this.
229. 11:56But I think, on some level, it might be a
230. 11:59little obvious for you.
231. 12:00So let's say we had-- let me get a good one here.
232. 12:07Let's say we had negative x to the third, plus 17x
233. 12:12squared, minus 70x.
234. 12:18Immediately, you say, gee, this isn't even a quadratic.
235. 12:20I don't know how to solve something like this.
236. 12:21It has an x to third power.
237. 12:23And the first thing you should realize is that every term
238. 12:26here is divisible by x.
239. 12:27So let's factor out an x.
240. 12:29Or even better, let's factor out a negative x.
241. 12:32So if you factor out a negative x, this is equal to
242. 12:35negative x times-- negative x to the third divided by
243. 12:38negative x is x squared.
244. 12:4117x squared divided by negative x is negative 17x.
245. 12:47Negative 70x divided by negative x is positive 70.
246. 12:52The x's cancel out.
247. 12:54And now, you have something that might look
248. 12:56a little bit familiar.
249. 12:59We have just a standard quadratic where the leading
250. 13:02coefficient is a 1.
251. 13:03We just have to find two numbers whose product is 70,
252. 13:08and that add up to negative 17.
253. 13:11And the numbers that immediately jumped into my
254. 13:12head are negative 10 and negative 7.
255. 13:17You take their product, you get 70.
256. 13:19You add them up, you get negative 17.
257. 13:21So this part right here is going to be x minus 10,
258. 13:25times x minus 7.
259. 13:28And of course, you have that leading negative x.
260. 13:31The general idea here is just see if there's anything you
261. 13:34can factor out.
262. 13:35And that'll get it into a form that you might recognize.
263. 13:37Hopefully, you found this helpful.
264. 13:38I want to reiterate what I showed you at the beginning of
265. 13:41this video.
266. 13:41I think it's a really cool trick, so to speak, to be able
267. 13:44to factor things that have a non-1 or non-negative 1
268. 13:49leading coefficient.
269. 13:50But to some degree, you're going to find out easier ways
270. 13:52to do this, especially with the quadratic
271. 13:54formula, in not too long.