1 00:00:00,000 --> 00:00:03,733 Let's try a rational roots 
example: so factor 2 00:00:03,733 --> 00:00:04,866 3 00:00:04,866 --> 00:00:08,333 this polynomial as a product 
of irreducible polynomials. 4 00:00:08,333 --> 00:00:09,600 5 00:00:09,600 --> 00:00:12,333 Actually there's a type of here I'm 
going to fix this typo and then come back. 6 00:00:12,333 --> 00:00:12,700 7 00:00:12,700 --> 00:00:13,500 8 00:00:13,500 --> 00:00:17,400 Okay turns out there was no 
typo, it's just my imagination, 9 00:00:17,400 --> 00:00:20,400 but let's factor this polynomial as a 
product of irreducible polynomials 10 00:00:20,400 --> 00:00:22,400 over the real numbers, R. 11 00:00:22,400 --> 00:00:25,400 So take a minute, try to 
see if you can factor this. 12 00:00:25,400 --> 00:00:28,633 Pause the video and then 
come back and hopefully 13 00:00:28,633 --> 00:00:31,433 you have a little bit of an 
idea of what's going on here. 14 00:00:31,433 --> 00:00:32,333 15 00:00:32,333 --> 00:00:34,700 The first thing I'd like to note 
is that we do have this 16 00:00:34,700 --> 00:00:37,100 denominator here 17 00:00:37,100 --> 00:00:38,700 and it's a little bit complicated to deal with. 18 00:00:38,700 --> 00:00:42,000 So what I'd like to do is I’d actually like to get 
rid of the denominator. Well how do I do that? 19 00:00:42,000 --> 00:00:44,266 I can common factor out... 20 00:00:44,266 --> 00:00:47,300 I can common factor out the denominator. 
So I'm going to common factor out 21 00:00:47,300 --> 00:00:49,266 1 over 15. 22 00:00:49,266 --> 00:00:52,733 So that'll leave me with a new 
polynomial with all integer coefficients. 23 00:00:52,733 --> 00:00:55,900 And why do I do this? Well once I do 
that, now I have an integral polynomial 24 00:00:55,900 --> 00:00:59,233 that I can now use the Rational 
Roots Theorem to try to factor. 25 00:00:59,233 --> 00:01:01,100 26 00:01:01,100 --> 00:01:04,766 So let's do that, we factor out the 
1 over 15 to get this polynomial. 27 00:01:04,766 --> 00:01:08,966 By the Rational Roots Theorem, we 
know that the numerator must divide 2, 28 00:01:08,966 --> 00:01:11,333 and the denominator must divide 15. 29 00:01:11,333 --> 00:01:14,233 So the factors of the denominator are 30 00:01:14,233 --> 00:01:16,466 plus minus 1, 3, 5, and 15, 31 00:01:16,466 --> 00:01:19,966 and the factors of 2 
are plus minus 1 or 2. 32 00:01:19,966 --> 00:01:20,800 33 00:01:20,800 --> 00:01:23,600 That leaves us with these 16 possible roots. 34 00:01:23,600 --> 00:01:26,666 That's a lot to try, but the way I would 35 00:01:26,666 --> 00:01:30,300 go about this is I would start 
by attempting the first… 36 00:01:30,300 --> 00:01:34,833 the integer ones. So try plus or 
minus 1 and try plus or minus 2. 37 00:01:34,833 --> 00:01:38,500 Sum of the coefficients isn’t 0 if 
I plug in minus 1, I see that I get 38 00:01:38,500 --> 00:01:43,133 negative 15 minus 32, that's already pretty 
negative and this is not going to cancel that 39 00:01:43,133 --> 00:01:45,033 So I know plus or minus 1 doesn't work. 40 00:01:45,033 --> 00:01:49,366 Let's try plus or minus 2. If 
try 2 here, I'm going to get… 41 00:01:49,366 --> 00:01:52,800 this one's going to be my 
biggest term, it's negative 128. 42 00:01:52,800 --> 00:01:57,100 That's 6 plus 2 that's 8, so I need 
this to be 120 if it's going to work. 43 00:01:57,100 --> 00:01:59,766 2 cubed is 8, 8 times 15 is 120, 44 00:01:59,766 --> 00:02:04,166 so 120 plus 8 is 128, these two 
cancel, I found a root, x equals 2. 45 00:02:04,166 --> 00:02:06,066 46 00:02:06,066 --> 00:02:08,766 So hence, by the Factor 
Theorem, x minus 2 is a factor. 47 00:02:08,766 --> 00:02:10,633 Now you have two options at this point: 48 00:02:10,633 --> 00:02:13,266 you could keep trying rational roots, 49 00:02:13,266 --> 00:02:17,200 or you can do the long division and I'm going 
to recommend here doing the long division. 50 00:02:17,200 --> 00:02:19,433 Why do I recommend doing that? 51 00:02:19,433 --> 00:02:22,533 The reason why is that if I do 52 00:02:22,533 --> 00:02:26,200 the long division here, I know that 
I'm going to be left with a quadratic, 53 00:02:26,200 --> 00:02:29,966 and I can factor quadratics. I have the 
quadratic formula so in the worst case, 54 00:02:29,966 --> 00:02:32,733 I can always factor a quadratic polynomial. 55 00:02:32,733 --> 00:02:35,233 So I have no problems actually 
doing the long division here 56 00:02:35,233 --> 00:02:38,433 because I know if I do that, I'm going 
to get to something that I can solve quickly. 57 00:02:38,433 --> 00:02:39,466 58 00:02:39,466 --> 00:02:42,933 So do the long division, just 
roll up your sleeves and do it. 59 00:02:42,933 --> 00:02:44,933 60 00:02:44,933 --> 00:02:49,366 x minus 2 into 15x cubed minus 
32x squared plus 3x plus 2. 61 00:02:49,366 --> 00:02:52,600 So x goes into 15x cubed, 
15x squared times. 62 00:02:52,600 --> 00:02:56,200 Multiply 15x squared by 
x minus 2, I get this. 63 00:02:56,200 --> 00:02:59,233 Take the difference, I get minus 2x squared. 64 00:02:59,233 --> 00:03:03,566 x goes into minus 2x squared 
minus 2x times, so on and so forth… 65 00:03:03,566 --> 00:03:05,633 66 00:03:05,633 --> 00:03:07,633 and keep going. 67 00:03:07,633 --> 00:03:08,700 68 00:03:08,700 --> 00:03:12,533 That gives me 15x squared minus 2x 
minus 1. I know my remainder here is 0, 69 00:03:12,533 --> 00:03:15,000 I didn't actually write the last 
line because I don't really need to, 70 00:03:15,000 --> 00:03:18,200 right, I know that 1 times this 
is going to give me exactly this, 71 00:03:18,200 --> 00:03:20,433 and it's supposed to 
because I know it's a factor. 72 00:03:20,433 --> 00:03:22,800 So there we go, there's my quotient. 73 00:03:22,800 --> 00:03:25,766 So if I factor this, I get 1 over 15 times 74 00:03:25,766 --> 00:03:29,666 x minus 2, times 15x 
squared minus 2x minus 1. 75 00:03:29,666 --> 00:03:33,733 Now this is a quadratic polynomial, worst 
case I plug it into the quadratic formula. 76 00:03:33,733 --> 00:03:36,100 I'm going to try to… 77 00:03:36,100 --> 00:03:38,633 the first thing I'm going to try to do 
though is I'm always going to try to factor, 78 00:03:38,633 --> 00:03:41,533 right? If I can actually 
factor this, then I'm done. 79 00:03:41,533 --> 00:03:42,566 80 00:03:42,566 --> 00:03:44,600 1 doesn't have too many 
roots, it's got to be 81 00:03:44,600 --> 00:03:46,633 plus 1 and minus 1 in some order. 82 00:03:46,633 --> 00:03:49,866 15, it could be 1 or 15, but that's 
not going to give me minus 2. 83 00:03:49,866 --> 00:03:53,500 5 and 3 in some combinations 
should give me minus 2, so if I put 84 00:03:53,500 --> 00:03:56,300 5x and 3x, I should be able to 
multiply them together to get 15, 85 00:03:56,300 --> 00:04:00,900 and I get the minus 2 in the middle because 
it's 5x times minus 1 and 3x times 1. 86 00:04:00,900 --> 00:04:02,066 87 00:04:02,066 --> 00:04:03,933 Beautiful, okay. 88 00:04:03,933 --> 00:04:07,800 So I factored this without too much 
stress at the end, and that's it. 89 00:04:07,800 --> 00:04:10,733 Once I have this factorization I'm done. 
So notice that, a couple of things, 90 00:04:10,733 --> 00:04:13,566 you could have found 
the rational roots 1/3, 91 00:04:13,566 --> 00:04:16,700 or you could have found the 
rational roots negative 1/5 above, 92 00:04:16,700 --> 00:04:18,933 and if you've done that you would have… 93 00:04:18,933 --> 00:04:22,833 you could have factored everything quickly if you 
found all three using the Rational Root Theorem, 94 00:04:22,833 --> 00:04:23,400 95 00:04:23,400 --> 00:04:26,366 or if you could have found - if you found one 
of these other ones, then you could have 96 00:04:26,366 --> 00:04:29,366 used long division and then you would 
have to factor the remaining quadratic, 97 00:04:29,366 --> 00:04:32,000 which is perfectly fine, either way is fine. 98 00:04:32,000 --> 00:04:34,600 This just gives you an 
example of how to do this. 99 00:04:34,600 --> 00:04:36,500 100 00:04:36,500 --> 00:04:38,766 A conjugate roots example, so how 
about something a little bit different? 101 00:04:38,766 --> 00:04:41,333 So factor the following polynomial 102 00:04:41,333 --> 00:04:43,533 over the complex numbers 103 00:04:43,533 --> 00:04:45,500 into a product of irreducible polynomials 104 00:04:45,500 --> 00:04:48,100 given that 2 minus 3i is a root. 105 00:04:48,100 --> 00:04:50,500 So here the problem’s a little 
bit different. Instead of telling you 106 00:04:50,500 --> 00:04:52,500 nothing, we actually tell you a root 107 00:04:52,500 --> 00:04:55,500 and say, “Hey, can you find the other ones?” 108 00:04:55,500 --> 00:04:57,666 Pause the video, give it a 
shot. Try to see if you can 109 00:04:57,666 --> 00:04:59,966 at least come up with a couple of roots 110 00:04:59,966 --> 00:05:02,033 and see if you can simplify this. 111 00:05:02,033 --> 00:05:03,300 112 00:05:03,300 --> 00:05:05,666 Okay, hopefully you've given this a shot. 113 00:05:05,666 --> 00:05:06,666 114 00:05:06,666 --> 00:05:09,233 One thing to note that when 
we have a real polynomial, 115 00:05:09,233 --> 00:05:13,566 so this polynomial is real, and we have a 
complex root, strictly complex root, 2 minus 3i, 116 00:05:13,566 --> 00:05:15,366 we know that 117 00:05:15,366 --> 00:05:18,533 this is one root and another 
root is given by its conjugate. 118 00:05:18,533 --> 00:05:20,766 That's the Conjugate Roots Theorem, okay 119 00:05:20,766 --> 00:05:21,933 120 00:05:21,933 --> 00:05:26,266 So by the Factor Theorem, we know that 
z minus those two roots must be factors, 121 00:05:26,266 --> 00:05:29,500 we multiply them together and we 
get a quadratic factor of this quartic. 122 00:05:29,500 --> 00:05:33,033 So if I do the long division and 
divide this quartic by the quadratic, 123 00:05:33,033 --> 00:05:36,000 then I'm left with another quadratic, 
and I know I can factor those. 124 00:05:36,000 --> 00:05:37,233 125 00:05:37,233 --> 00:05:40,100 Notice that this is over C, so the 
Fundamental Theorem of Algebra, 126 00:05:40,100 --> 00:05:42,466 or CPN, whichever theorem you want to use 127 00:05:42,466 --> 00:05:46,033 tells us that this will factor as a 
product of four linear things 128 00:05:46,033 --> 00:05:46,666 129 00:05:46,666 --> 00:05:49,300 Four linear terms… 130 00:05:49,300 --> 00:05:52,966 products…elements… polynomials, there we go. 131 00:05:52,966 --> 00:05:54,600 132 00:05:54,600 --> 00:05:56,400 Okay, here we go. 133 00:05:56,400 --> 00:05:57,866 134 00:05:57,866 --> 00:06:00,033 So we do that, we find 
our quadratic factor, 135 00:06:00,033 --> 00:06:02,966 z minus 2 minus 3i 
and z minus 2 plus 3i, 136 00:06:02,966 --> 00:06:05,633 that gives us z squared minus 4z plus 13. 137 00:06:05,633 --> 00:06:07,800 We do the long division, I didn't do it 138 00:06:07,800 --> 00:06:10,700 in this slide. You should do it, make 
sure that you know how to do this. 139 00:06:10,700 --> 00:06:12,700 We already saw one example, 
there’s two on the website. 140 00:06:12,700 --> 00:06:15,066 I don't feel like I need to do many more. 141 00:06:15,066 --> 00:06:18,200 That gives us the following though: 
z squared minus 4z plus 13, 142 00:06:18,200 --> 00:06:21,266 times z squared minus z minus 1. 143 00:06:21,266 --> 00:06:24,800 This last polynomial doesn't factor, but 
we can use the quadratic formula, 144 00:06:24,800 --> 00:06:26,700 right? We have a quadratic, 145 00:06:26,700 --> 00:06:29,366 we know that we can find the roots 
by using the quadratic formula. 146 00:06:29,366 --> 00:06:33,000 Plug it in, simplify, we get 1 
plus or minus root 5 over 2, 147 00:06:33,000 --> 00:06:33,833 148 00:06:33,833 --> 00:06:36,133 and if we simplify 
this what do we get? 149 00:06:36,133 --> 00:06:37,033 150 00:06:37,033 --> 00:06:40,266 Okay, so sorry, we have [these] 
roots, this is all there is to it. 151 00:06:40,266 --> 00:06:43,766 So now we have two more roots, we have the 
two roots from before, we know all the factors, 152 00:06:43,766 --> 00:06:46,300 right, just takes z minus 
root 1, z minus root 2, 153 00:06:46,300 --> 00:06:49,000 z minus root 3, and 
z minus the 4th root. 154 00:06:49,000 --> 00:06:50,433 And that's it. 155 00:06:50,433 --> 00:06:53,033 That gives us the 
factorization of this polynomial. 156 00:06:53,033 --> 00:06:54,133 157 00:06:54,133 --> 00:06:56,466 So there's an example 
with the conjugate roots. 158 00:06:56,466 --> 00:06:58,133 159 00:06:58,133 --> 00:07:00,566 Again so if you're given a root 
you have a little bit of information, 160 00:07:00,566 --> 00:07:04,400 use that to your advantage. Try to find another 
root using, let's say, the Conjugate Roots Theorem 161 00:07:04,400 --> 00:07:06,933 or Rational Roots Theorem, something like this. 162 00:07:06,933 --> 00:07:08,800 Worst case, you divide 
out and hope for the best, 163 00:07:08,800 --> 00:07:11,733 but hopefully you don't 
divide out by z minus... 164 00:07:11,733 --> 00:07:12,500 165 00:07:12,500 --> 00:07:15,400 hopefully you don't just divide out by this 
term because it's going to be a little bit ugly 166 00:07:15,400 --> 00:07:17,400 and you're going to have a lot of 
complex numbers floating around. 167 00:07:17,400 --> 00:07:19,100 Because you divided out by the product, 168 00:07:19,100 --> 00:07:21,500 we actually ended up with a 
very, very nice term at the end. 169 00:07:21,500 --> 00:07:23,133 So something to think about.