1 00:00:00,000 --> 00:00:00,450 2 00:00:00,450 --> 00:00:03,510 Divide and express as a simplified rational. 3 00:00:03,509 --> 00:00:05,089 State the domain. 4 00:00:05,089 --> 00:00:06,580 We start off with this expression. 5 00:00:06,580 --> 00:00:09,759 We actually have one rational expression divided by another 6 00:00:09,759 --> 00:00:10,959 rational expression. 7 00:00:10,960 --> 00:00:13,850 And like we've seen multiple times before, these rational 8 00:00:13,849 --> 00:00:16,640 expressions aren't defined when their denominators are 9 00:00:16,640 --> 00:00:17,920 equal to 0. 10 00:00:17,920 --> 00:00:23,150 So p plus 5 cannot be equal to 0, or if we subtract both 11 00:00:23,149 --> 00:00:26,070 sides of this-- we can't call it an equation, but we could 12 00:00:26,070 --> 00:00:29,070 call it a not-equation-- by negative 5, if we subtract 13 00:00:29,070 --> 00:00:34,399 negative 5 from both sides, you get p cannot be equal to-- 14 00:00:34,399 --> 00:00:36,379 these cancel out-- negative 5. 15 00:00:36,380 --> 00:00:38,010 That's what that tells us. 16 00:00:38,009 --> 00:00:41,129 Over here, we could do the same exercise for p plus 20 17 00:00:41,130 --> 00:00:43,060 also cannot be equal to 0. 18 00:00:43,060 --> 00:00:46,390 If it was, this expression would be undefined. 19 00:00:46,390 --> 00:00:48,299 Subtract 20 from both sides. 20 00:00:48,299 --> 00:00:50,699 4p cannot be equal to negative 20. 21 00:00:50,700 --> 00:00:52,330 Divide both sides by 4. 22 00:00:52,329 --> 00:00:54,409 p cannot be equal to negative 5. 23 00:00:54,409 --> 00:00:57,859 So in both situations, p being equal to negative 5 would make 24 00:00:57,859 --> 00:01:02,840 either of these rational expressions undefined. 25 00:01:02,840 --> 00:01:16,700 So the domain here is the set of all reals such-- or p is 26 00:01:16,700 --> 00:01:22,010 equal to the set of all reals such that p does not equal 27 00:01:22,010 --> 00:01:24,870 negative 5, or essentially all numbers except for negative 5, 28 00:01:24,870 --> 00:01:25,800 all real numbers. 29 00:01:25,799 --> 00:01:29,259 We've stated the domain, so now let's actually simplify 30 00:01:29,260 --> 00:01:30,870 this expression. 31 00:01:30,870 --> 00:01:34,160 When you divide by a fraction or a rational expression, it's 32 00:01:34,159 --> 00:01:36,319 the same thing as multiplying by the inverse. 33 00:01:36,319 --> 00:01:39,359 Let me just rewrite this thing over here. 34 00:01:39,359 --> 00:01:47,390 2p plus 6 over p plus 5 divided by 10 over 4p plus 20 35 00:01:47,390 --> 00:01:51,049 is the same thing as multiplying by the reciprocal 36 00:01:51,049 --> 00:01:56,840 here, multiplying by 4p plus 20 over 10. 37 00:01:56,840 --> 00:01:59,469 I changed the division into a multiplication and I flipped 38 00:01:59,469 --> 00:02:00,870 this guy right here. 39 00:02:00,870 --> 00:02:05,530 Now, this is going to be equal to 2p plus 6 times 4p plus 20 40 00:02:05,530 --> 00:02:07,349 in the numerator. 41 00:02:07,349 --> 00:02:08,769 I won't skip too many steps. 42 00:02:08,770 --> 00:02:09,419 Let me just write that. 43 00:02:09,419 --> 00:02:16,480 2p plus 6 times 4p plus 20 in the numerator and then p plus 44 00:02:16,479 --> 00:02:21,069 5 times 10 in the denominator. 45 00:02:21,069 --> 00:02:22,939 Now, in order to see if we can simplify this, we need to 46 00:02:22,939 --> 00:02:25,689 completely factor all of the terms in the numerator and the 47 00:02:25,689 --> 00:02:26,849 denominator. 48 00:02:26,849 --> 00:02:30,590 In the numerator, 2p plus 6, we can factor out a 2, so the 49 00:02:30,590 --> 00:02:38,750 2p plus 6 we can rewrite it as 2 times p plus 3. 50 00:02:38,750 --> 00:02:43,240 Then the 4p plus 20, we can rewrite that. 51 00:02:43,240 --> 00:02:50,870 We can factor out a 4 as-- so 4 times p plus 5. 52 00:02:50,870 --> 00:02:54,030 Then we have our p plus 5 down there in the denominator. 53 00:02:54,030 --> 00:02:55,629 We have this p plus 5. 54 00:02:55,629 --> 00:02:57,280 We can just write it down in the denominator. 55 00:02:57,280 --> 00:03:00,229 56 00:03:00,229 --> 00:03:03,959 Even 10, we can factor that further into its prime 57 00:03:03,960 --> 00:03:06,540 components or into its prime factorization. 58 00:03:06,539 --> 00:03:12,709 We can factor 10 into 2 times 5. 59 00:03:12,710 --> 00:03:14,610 That's the same thing as 10. 60 00:03:14,610 --> 00:03:15,690 Let's see what we can simplify. 61 00:03:15,689 --> 00:03:18,689 Of course, this whole time, we have to add the caveat that p 62 00:03:18,689 --> 00:03:20,609 cannot equal negative 5. 63 00:03:20,610 --> 00:03:22,540 We have to add this restriction on the domain in 64 00:03:22,539 --> 00:03:25,239 order for it to be the same expression as the one we 65 00:03:25,240 --> 00:03:26,730 started off with. 66 00:03:26,729 --> 00:03:27,679 Now, what can we cancel out? 67 00:03:27,680 --> 00:03:29,240 We have a 2 divided by a 2. 68 00:03:29,240 --> 00:03:30,330 Those cancel out. 69 00:03:30,330 --> 00:03:32,550 We have a p plus 5 divided by a p plus 5. 70 00:03:32,550 --> 00:03:34,800 We know that p plus 5 isn't going to be equal to 0 because 71 00:03:34,800 --> 00:03:37,910 of this constraint, so we can cancel those out. 72 00:03:37,909 --> 00:03:39,049 What are we left with? 73 00:03:39,050 --> 00:03:46,980 In the numerator, we have 4 times p plus 3, and in the 74 00:03:46,979 --> 00:03:52,609 denominator, all we have is that green 5, and we're done! 75 00:03:52,610 --> 00:03:55,550 We could right this as 4/5 times p plus 3, or just the 76 00:03:55,550 --> 00:03:56,719 way we did it right there. 77 00:03:56,719 --> 00:03:59,060 But we don't want to forget that we have to add the 78 00:03:59,060 --> 00:04:03,259 constraint p cannot be equal to negative 5, so that this 79 00:04:03,259 --> 00:04:06,979 thing is mathematically equivalent to 80 00:04:06,979 --> 00:04:09,639 this thing right here. 81 00:04:09,639 --> 00:04:09,865