1 00:00:00,400 --> 00:00:03,333 Hello everyone. In this video, we continue our 2 00:00:03,333 --> 00:00:06,933 voyage through the Fall 2000 exam, 3 00:00:06,933 --> 00:00:09,200 and now we've reached question 7. So 4 00:00:09,200 --> 00:00:12,000 part 1 is state the Rational Roots Theorem. 5 00:00:12,000 --> 00:00:12,866 6 00:00:12,866 --> 00:00:15,166 First part’s actually not too, too bad. 7 00:00:15,166 --> 00:00:16,166 8 00:00:16,166 --> 00:00:18,600 Now of course, on our exam, 
we won't have this issue 9 00:00:18,600 --> 00:00:20,366 because we'll be given all the theorems, 10 00:00:20,366 --> 00:00:25,166 but let's state it anyways. So let f of x 
be a polynomial with integer coefficients. 11 00:00:25,166 --> 00:00:28,133 If r equals a over b is a rational root of f of x, then 12 00:00:28,133 --> 00:00:30,066 the numerator must divide a naught 13 00:00:30,066 --> 00:00:32,633 and the denominator must divide 
the leading coefficient a n. 14 00:00:32,633 --> 00:00:34,366 15 00:00:34,366 --> 00:00:37,900 So clearly this is going to be a question about the 
Rational Roots Theorem based on the first part. 16 00:00:37,900 --> 00:00:40,100 Part 2: let f of x be the polynomial 17 00:00:40,100 --> 00:00:43,900 4x to the power of 4 plus 8x cubed 
plus 9x squared plus 5x plus 1. 18 00:00:43,900 --> 00:00:47,066 Determine the rational roots of f of x. 19 00:00:47,066 --> 00:00:48,633 Okay. 20 00:00:48,633 --> 00:00:51,333 So we'd like to determine the rational roots of f of x. So 21 00:00:51,333 --> 00:00:55,433 by the previous theorem we know 
that the only possibilities are 22 00:00:55,433 --> 00:00:59,100 when a divisor of 1 is in the numerator, 
and a divisor of 4 is in the denominator. 23 00:00:59,100 --> 00:01:03,633 This gives us 6 possibilities: plus minus 1, 
plus minus a half, and plus minus a quarter. 24 00:01:03,633 --> 00:01:04,600 25 00:01:04,600 --> 00:01:08,000 Now in order to save time, and as you 
can see this question is very lengthy, 26 00:01:08,000 --> 00:01:11,866 I'm just going to plug them in and go with it. 27 00:01:11,866 --> 00:01:14,466 So f at 1 gives me 27, 
f at minus 1 gives me 1, 28 00:01:14,466 --> 00:01:16,866 f at a half gives me 7, f at negative a half gives me 0, 29 00:01:16,866 --> 00:01:20,000 f at a quarter is 189 over 64, 30 00:01:20,000 --> 00:01:24,500 and f at negative 1 quarter 
is 13 over 64, okay? 31 00:01:24,500 --> 00:01:26,133 Only one of these is 0, 32 00:01:26,133 --> 00:01:28,800 that's the most important 
part, f at negative 1 half, 33 00:01:28,800 --> 00:01:32,366 so negative 1 half is the only rational root of f of x. 34 00:01:32,366 --> 00:01:33,666 35 00:01:33,666 --> 00:01:37,100 The final question asks us to factor 
this over four different rings 36 00:01:37,100 --> 00:01:39,400 or four different polynomial rings. So over 37 00:01:39,400 --> 00:01:42,466 Q of x, R of x, C of x, and Z 5 of x, okay? 38 00:01:42,466 --> 00:01:43,700 39 00:01:43,700 --> 00:01:48,700 So clearly we want to use the fact that we 
have a rational root, namely negative 1 half. 40 00:01:48,700 --> 00:01:51,733 So by the Factor Theorem, we know 
that if negative 1 half is a rational root, 41 00:01:51,733 --> 00:01:54,166 2x plus 1 must be a factor. 42 00:01:54,166 --> 00:01:56,733 So if we know a factor, 
we can just divide out 43 00:01:56,733 --> 00:01:59,400 by using long division, 
and get our other factor. 44 00:01:59,400 --> 00:02:00,000 45 00:02:00,000 --> 00:02:02,833 So let's do our long division, here I've done it in 46 00:02:02,833 --> 00:02:04,633 some paint program. 47 00:02:04,633 --> 00:02:09,566 So 2x plus 1 goes into 4x to the power of 4 
plus 8x cubed plus 9x squared plus 5x plus 1. 48 00:02:09,566 --> 00:02:13,100 How do we do this? Well compare 
leading coefficients, so 2x into 49 00:02:13,100 --> 00:02:15,766 4x to the 4, that's 2x cubed. 50 00:02:15,766 --> 00:02:18,800 So we do the division, and what do we get? 51 00:02:18,800 --> 00:02:23,066 We're going to get - so we 
multiply 2x plus 1 by 2x cubed, 52 00:02:23,066 --> 00:02:25,266 that gives us 4x to the 
power of 4 plus 2x cubed. 53 00:02:25,266 --> 00:02:27,366 Subtract we get 6x cubed. 54 00:02:27,366 --> 00:02:29,166 Drop down the other term, 9x squared. 55 00:02:29,166 --> 00:02:32,366 2x goes into 6x cubed 3x squared times. 56 00:02:32,366 --> 00:02:34,300 Write it down. 57 00:02:34,300 --> 00:02:34,833 58 00:02:34,833 --> 00:02:39,400 Okay then simplify, so 9 minus 3 gives us the 6x squared, drop down the 5. 59 00:02:39,400 --> 00:02:43,466 2x goes into 6x squared 3x 
times. 6x squared plus 3x, 60 00:02:43,466 --> 00:02:48,533 multiply that through, we're going to get 2x 
plus 1, that's good, we want a remainder 0. 61 00:02:48,533 --> 00:02:51,233 Now when we do this though we 
should keep note of something, right? 62 00:02:51,233 --> 00:02:53,600 So we do this and we have 63 00:02:53,600 --> 00:02:57,333 that f of x - I'm keeping this hidden for 
a reason - f of x is equal to 2x plus 1 64 00:02:57,333 --> 00:03:01,700 times 2x cubed plus 3x squared plus 
3x plus 1, that's the Division Algorithm. 65 00:03:01,700 --> 00:03:02,566 66 00:03:02,566 --> 00:03:05,633 Now I've always said that the goal of these types of factoring questions 67 00:03:05,633 --> 00:03:09,600 is to get down to a quadratic. When you get down to a quadratic, you know what to do. 68 00:03:09,600 --> 00:03:12,400 We unfortunately still have a cubic, 69 00:03:12,400 --> 00:03:15,200 so if we have a cubic and 
we still need to factor it, 70 00:03:15,200 --> 00:03:17,600 well we really don't have too many tricks, right? They didn't 71 00:03:17,600 --> 00:03:19,866 give us a factor in this question, 72 00:03:19,866 --> 00:03:22,833 so we're left with only one option 73 00:03:22,833 --> 00:03:25,766 and our one option to attempt to factor this, 74 00:03:25,766 --> 00:03:30,733 and pray to God this is going to work, is that 
this thing has another factor that's rational, 75 00:03:30,733 --> 00:03:34,733 but we already saw that there's only 
one possibility for a rational factor, 76 00:03:34,733 --> 00:03:36,333 and that's negative a half. 77 00:03:36,333 --> 00:03:40,233 So negative a half might be a repeated 
root, that's our only hope at this point. 78 00:03:40,233 --> 00:03:43,400 Let's give it a shot and plug 
negative a half into this cubic. 79 00:03:43,400 --> 00:03:44,366 80 00:03:44,366 --> 00:03:47,166 This is my comment here: we close 
our eyes and hope that this is a root. 81 00:03:47,166 --> 00:03:50,666 So this I actually decided to plug 
in, so plug in negative a half, 82 00:03:50,666 --> 00:03:53,466 simplify, we actually get 0. 83 00:03:53,466 --> 00:03:56,600 2x plus 1 is a repeated factor of f of x. 84 00:03:56,600 --> 00:03:57,266 85 00:03:57,266 --> 00:03:59,466 So let's do another long division. 86 00:03:59,466 --> 00:04:00,800 87 00:04:00,800 --> 00:04:04,266 So if we do our long division, 
2x into 2x cubed is x squared, 88 00:04:04,266 --> 00:04:07,666 simplify we get 2x squared plus 3x. Again x 89 00:04:07,666 --> 00:04:12,866 times 2x gives me 2x squared. 
Do that, simplify, again we get 0. 90 00:04:12,866 --> 00:04:14,900 So now our factorization is 91 00:04:14,900 --> 00:04:19,300 f of x equals 2x plus 1 squared 
times x squared plus x plus 1. 92 00:04:19,300 --> 00:04:21,766 It should be noted if I expand this out, this is 93 00:04:21,766 --> 00:04:25,200 going to also work over Z mod 5, 94 00:04:25,200 --> 00:04:27,966 which is another field 
that have to work over so 95 00:04:27,966 --> 00:04:31,733 keep in mind this factorization is good 
over all four of those rings, okay? 96 00:04:31,733 --> 00:04:33,833 97 00:04:33,833 --> 00:04:36,633 So the roots of this last equation are 
given by the quadratic formula, right, 98 00:04:36,633 --> 00:04:40,000 I mean, so now we have a quadratic, we know 
what to do. Plug in the quadratic formula, 99 00:04:40,000 --> 00:04:44,266 and when we plug in we get minus 
1 plus or minus root 3i over 2. 100 00:04:44,266 --> 00:04:44,900 101 00:04:44,900 --> 00:04:46,800 These factors are imaginary, 102 00:04:46,800 --> 00:04:49,666 so we know we have the 
complete factorization over Q 103 00:04:49,666 --> 00:04:53,000 and the complete factorization 
over R, it's given by this, 104 00:04:53,000 --> 00:04:56,100 and to get the factorization over C, 
well these are our two roots so just 105 00:04:56,100 --> 00:04:58,900 split them up and write 
them as factors, okay? 106 00:04:58,900 --> 00:05:00,000 107 00:05:00,000 --> 00:05:02,200 Fantastic. 108 00:05:02,200 --> 00:05:05,866 So now over Z 5, we have 
to do one little check, right? 109 00:05:05,866 --> 00:05:09,500 So now we have a quadratic polynomial and 
we want to know if it's irreducible over Z 5. 110 00:05:09,500 --> 00:05:14,033 Well from the last homework question, the 
last homework of this course, question 7, 111 00:05:14,033 --> 00:05:17,400 tells us that if this thing factors, 
it must have a root, right? 112 00:05:17,400 --> 00:05:18,333 113 00:05:18,333 --> 00:05:21,466 If it's going to factor as something, it's got 
to factor as a linear term times a linear term 114 00:05:21,466 --> 00:05:25,500 to get to a quadratic term. So if there's 
a linear factor, then it must have a root 115 00:05:25,500 --> 00:05:28,966 in Z 5. So just check the five 
numbers and see if it has a root. 116 00:05:28,966 --> 00:05:33,633 So we plug in 0, 1, 2, 3, 4, we 
clearly don't get 0 in either case, 117 00:05:33,633 --> 00:05:36,133 and we're done. So thus this has 
no roots, and hence the factorization 118 00:05:36,133 --> 00:05:39,066 over Z 5 is the same over Q and as over R. 119 00:05:39,066 --> 00:05:40,333 120 00:05:40,333 --> 00:05:43,633 Pretty crazy question, there's a ton 
of work that went into this question. 121 00:05:43,633 --> 00:05:46,366 If you notice, I think this took 
three pages, but that's okay. 122 00:05:46,366 --> 00:05:49,766 As long - I mean, you're making 
progress towards the goal. 123 00:05:49,766 --> 00:05:53,100 It's a tougher question, it’s a longer 
question, it's a lot of parts, but that's okay. 124 00:05:53,100 --> 00:05:55,466 Again, keep at it, don't give up, 125 00:05:55,466 --> 00:05:56,500 126 00:05:56,500 --> 00:06:01,466 and you have to notice a couple of things to hit 
the Graceland, you know, to get there, but 127 00:06:01,466 --> 00:06:04,733 it can be done, okay, that's 
kind of the point that I'm making. 128 00:06:04,733 --> 00:06:07,833 Again, trickier question, a little 
harder than the previous ones, but 129 00:06:07,833 --> 00:06:10,266 it can still be done. 130 00:06:10,266 --> 00:06:14,633 It helps to read questions beforehand it's sort of a good tip in any exam. 131 00:06:14,633 --> 00:06:18,266 Reading questions beforehand definitely helps and I hope that this video helps 132 00:06:18,266 --> 00:06:22,166 solve this problem, give you a little bit of insight 
as what to do when you're trying to factor things. 133 00:06:22,166 --> 00:06:23,499 Good luck.