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College Algebra: Rational Zero Theorem


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About this Lesson

  • Type: Video Tutorial
  • Length: 7:15
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 78 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: The Rational Root Theorem (2 lessons, $2.97)

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.

About this Author

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So what I want to start thinking about now is what's the situation if we know that a particular polynomial has rational zeros? Now, what's a rational number again? Well, a rational number is just a fraction, like or 7 or 15, a number that's not rational is a number that has like square roots in it and stuff like that, like the square root of 2, for example, is not a rational number. Well, it turns out that if you have rational zeros, or if you have rational roots, you might be able to make a list of all the possibilities where those rational roots or zeros, may live. So I'm going to tell you how to make a list out of all the possible candidates for rational zeros, and then you can check if you want to by hand and see if, in fact, any of them are really zeros.
And to inspire this, let me write down an example, and then we'll see the pattern. Suppose I look at the following polynomial: (3x - 1)--I'm going to give it to you in factored form--(x + 2)(2x - 5). You can multiply that out if you want, and you'd see it would be a cubic. But it's easier to write it this way, because if I were to ask you what are the zeros of this polynomial, it's a piece of cake. You just set that equal to zero. What would I see? What I would see is either this equals zero--I'm going to do this in my head now. See if you can sort of do it in your head too. If that equals zero, I bring the 1 over, that would be 3x = 1, so therefore x = 1/3. So I'd see x = 1/3. There's one zero. This zero is easier to see, it's x = -2, and this zero is okay to see if you're careful, it's going to be 5/2. So there are the three zeros for this cubic polynomial. Okay, so that's easy to see.
What if you were to multiply all this out, sort of do ultra-foiling? Foil this and then distribute all this. You'd get a whole big mess of stuff, right? Let's just see what the beginning of that mess would look like and what the end of the mess would look like. Now, look at this. I would see 3x times x times 2x, so I'd see at the beginning a 6x^3. And then I would see some other junk, right? Junk, junk, junk, junk, junk, junk, and then what would the last term be? -1 times 2 times -5, so that would be a +10.
Now if you think about it for a second, when I solved this for zero, I just used the 3 and the -1. When I solved this for zero I just used the 1 in front of here and the 2. When I solved this for zero I just used the 2 and the -5. So somehow the product of these three terms, being the first term, and the product of the three last terms, being the last term, that should somehow capture within it these answers. And how? Well, here's how you do it. What you do is you take a look at the coefficient on the highest power, and what you do is you write down all the possible factors of that number. So what are the factors of 6? Well, there's ± 1, because ± 1 divides into this evenly. ± 2, ± 3, and then ± 6. Those are all the numbers that divide evenly into 6, all the different kinds of factors, 1, 2, 3, and 6.
What about for 10? Let's now factor 10. If you factor 10 I see ± 1, ± 2, ± 5, and ± 10. Well, it turns out that if you're going to have fractions that will be zeros of your polynomial, then it will always be the case that the numerator of the fraction, that is to say, the top, always must be one of these numbers. And the denominator of that fraction that's going to be a zero must always be one of these numbers. So take a look. 1/3--Notice that that 1 came from here, and that 3 came from here. So, in fact, 1/3 came from a top from here and a bottom from here. What about the -2? The -2 we got by taking the -2 from here, and dividing it by the 1 here. You see I took a top over a bottom.
This 5/2, how did I get that? I took the 5 from here and I divided by the 2 from here. So basically, if you're looking at a polynomial, if you look at the coefficient of that leading term, the term that's in front of the highest power of x, all the factors of that will give you all the allowable terms for the denominator for any rational zero, and if you take a look at the constant number in your polynomial, in this case 10, what you would see is you factor all the allowable numerators for your rational roots.
So that means the following: Suppose someone just gave us the polynomial in this form and not in this hidden, this nice form? If someone said to you, "Are there rational roots?" All you have to look at are all the combinations of taking these kind of numbers divided by these numbers. So, in fact, I'll just verbally list them for you right now. They would be ± 1, 1/1, they would be ± , they would be ± 1/3, they would be ± 1/6. Do you see how I'm taking this hand and putting it over this hand? So all those things are candidates.
Then there's ± 2/1, ± 2/2, which is 1, we already saw that, ± 2/3, ± 2/6, which is ± 1/3. Then there's ± 5, ± 5/2, ± 5/3, ± 5/6. Then there's ± 10, ± 5, 10/2, ± 10/3, and ± 10/6. Those are the only possible candidates for rational zeros--that's it. And even though it's a big list, you'll notice it's a finite list. You can actually go through and check those things one after the other and see if any of those are roots. If you would have done that you would have found 1/3, - 2, and 5/2, which were some of the ones I mentioned, as the zeros. If you would have found none of them, if none of them would have been zeros, that means you had no rational roots. The roots either may be irrational, have square roots in them, or they may be imaginary.
We'll take a look at some very explicit examples up next, and we'll talk about this idea, which is really called "the rational zero theorem." I'll see you there.
Polynomial and Rational Functions
The Rational Root Theorem
Presenting the Rational Zero Theorem Page [2 of 2]

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