1 00:00:00,000 --> 00:00:03,900 Hello everyone. Welcome to Week 12 of Carmen's 
Core Concepts, my name is Carmen Bruni. 2 00:00:03,900 --> 00:00:07,133 In this video series, we've been talking about 3 00:00:07,133 --> 00:00:10,000 Math 135 on a week-by-week basis. 4 00:00:10,000 --> 00:00:12,933 This is the final week, Week 
12. This is a shorter video. 5 00:00:12,933 --> 00:00:15,300 Most of the content was in Week 11… 6 00:00:15,300 --> 00:00:16,100 7 00:00:16,100 --> 00:00:18,000 in the Week 11 videos. 8 00:00:18,000 --> 00:00:20,800 Okay, Real Quadratic Factors. 9 00:00:20,800 --> 00:00:23,933 What we'd like to - okay let's 
read this proposition first. 10 00:00:23,933 --> 00:00:26,433 Let f at x be a real polynomial. 11 00:00:26,433 --> 00:00:28,600 If c is a strictly complex root, 12 00:00:28,600 --> 00:00:33,200 so remember this is set difference so c is a 
complex number not inside the real numbers, 13 00:00:33,200 --> 00:00:35,533 and f at c is 0, so it's 
a root of our polynomial, 14 00:00:35,533 --> 00:00:39,400 then there exists a polynomial 
g of x, real, such that 15 00:00:39,400 --> 00:00:42,400 g of x is a real quadratic factor of f of x. 16 00:00:42,400 --> 00:00:44,800 17 00:00:44,800 --> 00:00:48,400 Okay so, again pause thevideo and try to prove this, 18 00:00:48,400 --> 00:00:51,933 and what you're going to 
possibly realize is that there are 19 00:00:51,933 --> 00:00:54,600 sort of two parts here. One part’s 
very subtle, which we'll talk about, 20 00:00:54,600 --> 00:00:56,533 and one part’s sort of clearer. 21 00:00:56,533 --> 00:00:59,100 22 00:00:59,100 --> 00:01:03,033 So if you're trying to prove this, again, 
we know from the previous videos 23 00:01:03,033 --> 00:01:04,900 that if c is a root 24 00:01:04,900 --> 00:01:08,200 of a real polynomial, then its 
conjugate is also a root. 25 00:01:08,200 --> 00:01:11,433 So if we use the fact that we have those 
two roots, we can actually pair them up 26 00:01:11,433 --> 00:01:15,400 and we're going to see that we 
get a quadratic real factor. 27 00:01:15,400 --> 00:01:20,033 So let's do that. Let's pair up 
c and its conjugate, c bar. 28 00:01:20,033 --> 00:01:23,300 We know that both of these are roots 
by the Conjugate Roots Theorem, 29 00:01:23,300 --> 00:01:24,533 30 00:01:24,533 --> 00:01:27,100 that will come into play a little bit later. 31 00:01:27,100 --> 00:01:29,600 If we multiply this out, we get 32 00:01:29,600 --> 00:01:34,600 the following: so we get c plus its 
conjugate, that's just 2 times the real part, 33 00:01:34,600 --> 00:01:38,700 and if we multiply c and its conjugate, 
we just get the modulus squared. 34 00:01:38,700 --> 00:01:42,233 That's Properties of Modulus, and 
that's Properties of Conjugates there. 35 00:01:42,233 --> 00:01:44,200 36 00:01:44,200 --> 00:01:48,800 So good, okay, so here's g of x. This is a 
real polynomial, it has c and c bar as a root, 37 00:01:48,800 --> 00:01:52,000 and we claim that g of 
x is a factor of f of x. 38 00:01:52,000 --> 00:01:54,300 Now here's where the 
subtle part comes in, 39 00:01:54,300 --> 00:01:56,200 and it turns out that it can't happen, 40 00:01:56,200 --> 00:01:58,300 but you do have to 
think about it a little bit. 41 00:01:58,300 --> 00:02:00,000 42 00:02:00,000 --> 00:02:01,833 What we talked about is 
we already know that 43 00:02:01,833 --> 00:02:06,333 x minus c and x minus c bar, we already 
know that these are both factors of f of x 44 00:02:06,333 --> 00:02:08,800 over the complex numbers,
that we know, right? 45 00:02:08,800 --> 00:02:10,666 We saw that before. 46 00:02:10,666 --> 00:02:13,500 The problem now is… 47 00:02:13,500 --> 00:02:17,133 if I take f of x and I divide it by 
x minus c and x minus c bar 48 00:02:17,133 --> 00:02:19,800 I know I get some quotient, 
let’s call it q of x, 49 00:02:19,800 --> 00:02:23,333 plus some 0 remainder, right, 
because they both divide it. 50 00:02:23,333 --> 00:02:24,200 51 00:02:24,200 --> 00:02:29,100 So at this point in our mind, we're 
thinking of that quotient, q of x, as being 52 00:02:29,100 --> 00:02:30,133 53 00:02:30,133 --> 00:02:32,300 a complex polynomial. 54 00:02:32,300 --> 00:02:32,900 55 00:02:32,900 --> 00:02:35,200 The question now becomes well why 56 00:02:35,200 --> 00:02:39,100 when I divide it by this 
polynomial, do I get 57 00:02:39,100 --> 00:02:42,500 a real polynomial with 0 
remainder over the reals? 58 00:02:42,500 --> 00:02:46,833 It's not clear that quotient 
actually is a real polynomial. 59 00:02:46,833 --> 00:02:49,333 It turns out that it is, which 
we'll see in a minute, 60 00:02:49,333 --> 00:02:52,533 but at this point it's not clear just 
because these two things are 61 00:02:52,533 --> 00:02:55,133 roots, and when I divide them 
over the complex numbers 62 00:02:55,133 --> 00:02:59,300 that I get some complex polynomial, it's not 
clear that if I divide by the product of these things, 63 00:02:59,300 --> 00:03:03,433 which is a real polynomial, that I get a 
real polynomial factor with no remainder. 64 00:03:03,433 --> 00:03:05,900 Again it's true, we'll prove it 
specifically in this case, 65 00:03:05,900 --> 00:03:08,600 you can do things a little bit 
more general than this, but 66 00:03:08,600 --> 00:03:11,400 let's just focus on this case for now. 67 00:03:11,400 --> 00:03:13,733 So last thing we have to do is 
show that g of x is a factor, 68 00:03:13,733 --> 00:03:16,500 how do we show that it’s a factor? We're 
going to use the Division Algorithm, 69 00:03:16,500 --> 00:03:20,300 and reach a contradiction by 
assuming that it's not a factor. 70 00:03:20,300 --> 00:03:21,900 71 00:03:21,900 --> 00:03:25,600 So by the Division Algorithm there 
are polynomials q of x and r of x, 72 00:03:25,600 --> 00:03:27,100 and these are unique, 73 00:03:27,100 --> 00:03:31,133 such that it satisfies this 
equation f is equal to g q plus r, 74 00:03:31,133 --> 00:03:34,100 and r is either 0 or its degree is less than 2. 75 00:03:34,100 --> 00:03:35,200 76 00:03:35,200 --> 00:03:39,100 So we're going to assume towards a 
contradiction that it's not the 0 polynomial, 77 00:03:39,100 --> 00:03:43,533 then the degree is either 0 or 1. That 
is, [r] of x is either linear or constant. 78 00:03:43,533 --> 00:03:47,733 When we plug in x equals c into this 
equation, what do we get? We see that 79 00:03:47,733 --> 00:03:51,400 well c was a root so we 
know that f at c is 0. 80 00:03:51,400 --> 00:03:55,600 c is the root of our 
quadratic polynomial g of c, 81 00:03:55,600 --> 00:03:59,733 so we know that's also 0. So 
we see that r of c must be 0. 82 00:03:59,733 --> 00:04:01,100 83 00:04:01,100 --> 00:04:03,700 So if r of x is constant, then 
we know it's the 0 constant 84 00:04:03,700 --> 00:04:06,533 because it's 0 at some point. 85 00:04:06,533 --> 00:04:09,433 Again this is actually also a little bit subtle, 
which I'm not going to talk about here, 86 00:04:09,433 --> 00:04:12,933 but this is constant at some 
complex point, right? 87 00:04:12,933 --> 00:04:16,133 So therefore this is the 0 polynomial 
as a complex polynomial, 88 00:04:16,133 --> 00:04:20,100 so why is it that the 0 polynomial’s a real 
polynomial? I'll let you think about that 89 00:04:20,100 --> 00:04:22,833 as an exercise, but I don’t 
want to talk about it too much. 90 00:04:22,833 --> 00:04:28,233 And if r of x was a linear polynomial, 
let's say r of x is equal to a x plus b, 91 00:04:28,233 --> 00:04:29,300 92 00:04:29,300 --> 00:04:31,133 then if we plug in c we get 93 00:04:31,133 --> 00:04:36,700 a times c plus b equals 0, and so c must 
be a real number, it's minus b over a, 94 00:04:36,700 --> 00:04:39,400 and that also is a contradiction. So 95 00:04:39,400 --> 00:04:42,500 if it's linear, then we see that 
our root must have been real, 96 00:04:42,500 --> 00:04:46,100 and if it's a constant polynomial, then we 
know that it must be the 0 polynomial. 97 00:04:46,100 --> 00:04:47,600 98 00:04:47,600 --> 00:04:49,700 So in either case, we see that 
the remainder must be 0, 99 00:04:49,700 --> 00:04:53,100 and hence g divides f, 
so g is a factor of f… 100 00:04:53,100 --> 00:04:55,200 is a real factor of f. 101 00:04:55,200 --> 00:04:56,833 102 00:04:56,833 --> 00:05:00,933 Okay good, so that satisfies 
the Real Quadratic Factors 103 00:05:00,933 --> 00:05:02,100 104 00:05:02,100 --> 00:05:03,800 question. Now we have 105 00:05:03,800 --> 00:05:08,700 the final part which is how 
do real polynomials factor? 106 00:05:08,700 --> 00:05:09,533 107 00:05:09,533 --> 00:05:12,200 And it turns out that real 
polynomials, they either factor as 108 00:05:12,200 --> 00:05:16,000 a product of linear factors, quadratic 
factors, or some combination of the two. 109 00:05:16,000 --> 00:05:17,200 110 00:05:17,200 --> 00:05:20,700 And that's what this theorem says, 
Real Factors of Real Polynomials. 111 00:05:20,700 --> 00:05:22,733 112 00:05:22,733 --> 00:05:25,900 My acronym is wrong here, it 
should be RFRP, not RFPF. 113 00:05:25,900 --> 00:05:26,900 114 00:05:26,900 --> 00:05:29,600 Okay, let's see a proof of this. 115 00:05:29,600 --> 00:05:32,900 I guess I should…before we read the 
proof, you should actually try this proof. 116 00:05:32,900 --> 00:05:35,733 It's a little bit… 117 00:05:35,733 --> 00:05:37,600 it's subtle, but you can do it. 118 00:05:37,600 --> 00:05:38,800 119 00:05:38,800 --> 00:05:41,000 And let me give you a hint, 
if you forget this proof, 120 00:05:41,000 --> 00:05:43,833 the way that we would 
start this proof is start by 121 00:05:43,833 --> 00:05:47,533 actually writing down all the possible 
roots, they're either strictly real, 122 00:05:47,533 --> 00:05:51,200 or they're strictly complex, and see if 
you can find some sort of relationship 123 00:05:51,200 --> 00:05:53,500 between factors and these roots. 124 00:05:53,500 --> 00:05:55,300 125 00:05:55,300 --> 00:05:57,800 So what's the idea here? This is a little bit 126 00:05:57,800 --> 00:05:59,800 of a sketchy proof, but… 127 00:05:59,800 --> 00:06:03,700 this is more of a proof sketch than 
a real formal proof, but that's okay. 128 00:06:03,700 --> 00:06:05,300 129 00:06:05,300 --> 00:06:08,433 So what do we do? We're going to take 
the real roots and the complex roots, 130 00:06:08,433 --> 00:06:12,933 and the complex roots come in pairs, one and 
its conjugate because this polynomial is real. 131 00:06:12,933 --> 00:06:16,600 For each pair, what you're going to do is you're
going to associate to it a quadratic factor, 132 00:06:16,600 --> 00:06:19,433 which we know from 
the previous theorem exists. 133 00:06:19,433 --> 00:06:23,033 So each pair of complex roots 
corresponds to a quadratic factor, 134 00:06:23,033 --> 00:06:27,300 and every individual real root 
corresponds to a real linear factor. 135 00:06:27,300 --> 00:06:28,000 136 00:06:28,000 --> 00:06:30,000 We put those two things together, 137 00:06:30,000 --> 00:06:31,133 138 00:06:31,133 --> 00:06:33,533 so here we have our quadratic 
factors and here our 139 00:06:33,533 --> 00:06:36,333 real linear factors are given by g1 to g k, 140 00:06:36,333 --> 00:06:41,133 multiply them together, and that gives us the 
f of x, up to some possible constant here, 141 00:06:41,133 --> 00:06:44,400 which is the - it turns out that this 
constant will be the leading term 142 00:06:44,400 --> 00:06:46,200 of our polynomial. 143 00:06:46,200 --> 00:06:47,700 144 00:06:47,700 --> 00:06:49,533 And that's it, so that's 
the main idea here, okay? 145 00:06:49,533 --> 00:06:53,100 So we can find the roots by CPN, 146 00:06:53,100 --> 00:06:55,700 Complex Polynomials Have n Roots, 147 00:06:55,700 --> 00:06:59,500 and once we have these roots, we can 
pair them up and match them to factors 148 00:06:59,500 --> 00:07:02,800 using Real Quadratic Factors, 
and using the Factor Theorem. 149 00:07:02,800 --> 00:07:04,200 150 00:07:04,200 --> 00:07:07,100 And that's it, so that's Real 
Factors of Real Polynomials. 151 00:07:07,100 --> 00:07:10,533 There's a couple more examples 
that we can talk about right now, 152 00:07:10,533 --> 00:07:12,500 but this is the last theorem of this course. 153 00:07:12,500 --> 00:07:16,000 How do I factor real polynomials? Well 
it factors as linear and quadratic factors.