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College Algebra: Using Descartes' Rule of Signs


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  • Type: Video Tutorial
  • Length: 6:50
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 73 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Polynomial & Rational Functions (23 lessons, $35.64)
College Algebra: Zeros of Polynomials (5 lessons, $7.92)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Okay, so we saw with the rational zero theorem a way of listing all the candidates for possible rational roots. But what if you just want to get a sense of the number of real roots and a sense of how many positive real roots there are and how many negative real roots there are? Well, it turns out there is this really clever, neat theorem, which really is nothing more than just a really clever, neat trick, and it's actually due to Descartes. Now, I know that a lot of people think of Descartes as sort of this philosopher kind of person. He said, "cogito ergo sum"--I think, therefore I am--but actually he's also a mathematician, and before he was thinking his first slogan was, "I factor, therefore I am." And in fact, this method that I'm about to show you, this really neat trick, is called Descartes' Rule of Sign. And here's the method. The method is that you write down your polynomial with the terms, in terms of exponents, decreasing. So you have the highest power of x followed immediately by the next highest power of x, followed immediately by the next highest power of x, and so on and so forth. So even if the polynomial is given to you in some sort of crazy order, you reorder it very carefully so that the exponents are in descending order. And then you look at sort of the sign changes. So let's do an example here and I'll show you exactly what it says and how it works.
So suppose I have a polynomial and it equals this: x^4[ ]- 6 x³ + 4x² + x - 5. Now, what do you do? You look at this thing, and now notice it's in descending order of exponents--4, 3, 2, 1, and then there's no exponent there at all. Now what you do is you take a look at the coefficient and just look at the sign changes. If there's a sign change, you count it. So, for example, this is a +1, that's a -6. so there's a change in sign. This is called Descartes' Rule of Signs, and so it's looking at the sign. Here I go from a -6 to a 4. That's a change in sign. Here I go from a 4 to a 1. That's not a change in sign, so I don't count that. But here I go from a 1 to a -5. This is a change in sign. So I count these up. So I have a 1, 1, 1. So there are three changes in sign. So you know what that means? It means that there are either going to be three real positive roots, or there will be an even number--this number minus an even number of roots. So, for example, there's either going to be three real roots, or if I take 2 away, there's going to be one positive real root.
So let me say the method again. You count how many changes of sign there are and there will be that many positive real roots or the number of positive real roots will be this number minus some even number. So, in this case there will either be three positive real roots, or, if I subtract 2, one positive real root. But that's it. Those are the only possibilities for positive real roots. So that's pretty cool--Descartes' Rule of Sign. All you do--count the number of sign changes and then the number of positive real roots will be this number or this number minus some even number. In this case, since 3 is so small, either there are three positive real roots or there are going to be 3 - 2 positive real roots. There can't be 3 - 4 positive real roots, because that would be a negative number, so that's out. Okay, what about for the negative real roots? Well, you do the exact same procedure, but you do it with the function f(-x). So if you look at -x for x in here--so plug in -x wherever you see an x--then what do you see? Well, if I put in a -x here and raise to the 4^th power, it becomes a plus x because of even exponents, so I have x^4. However, here, when I put in -x and cube it, I actually get -x³, because the negative is in front, so that makes a +6x³. When I put a -x in here and square it it becomes positive, so I have a +4x², and then when I put in this -x for x, I see a -x, -5.
Okay, great. Now I look for the change in signs. This is now f(-x). Remember, I replaced all the x's by -x's. I see no change in sign here--1 to 6, no change in sign here--6 to 4, I see a change in sign here from 4 to -1, there's a change in sign, but no change in sign from negative to negative. So the number of negative real roots will be either the number of changes of signs of f(-x) or that number minus some even number. So in this case since there's only one, I see there will definitely be one real negative zero, because there's only one possibility.
Here, how many positive real zeros are there? There's either going to be three or one, and we just don't know. But you can reduce the possibilities down. Suppose, for example, the number of changes in signs we had were five. If we had five changes in signs for this one right here, that means that the number of negative real zeros would either be five or three or one, because I just take the number, and it's either that many, or it's that many minus an even number. So it's either -2 or -4. So if I had five changes in signs in some polynomial for this, I would see the number of negative real roots would either be five, three, or one. Neat! So you can actually get a sense of how many positive real roots, how many negative real roots, or at least roughly how many there are by using Descartes' Rule of Signs. And always remember, folks, he first factored before he was. Enjoy.
Polynomial and Rational Functions
Zeros of Polynomials
Using Descartes' Rule of Sign Page [1 of 1]

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