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College Algebra: Polynomial Zeros & Multiplicities


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:09
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 87 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra Review (30 lessons, $59.40)
College Algebra: Polynomial & Rational Functions (23 lessons, $35.64)
College Algebra: Zeros of Polynomials (5 lessons, $7.92)

The zeros of a polynomial are just the places where a polynomial crosses the x-axis, or those values for x, which if you plug into the polynomial, give zero as a result. In this lesson, we will define and discuss zeros and their multiplicity for a variety of different functions from the perspective of how to identify and count zeros using a mathematical formula or using a graph of the function. We'll cover zeros and multiplicity for parabolas and various formulas like (x^2-4x+4), (7x^3+x), and [(x+1)^2*(x-1)^3*(x^2-10)]

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

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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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

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Recent Reviews

Prof Burger ROCKS!!!!
~ cbrown

I use these videos to help me teach my Alg I students. It is great to provide them with two teachers' point of views. Thanks Prof Burger!

Prof Burger ROCKS!!!!
~ cbrown

I use these videos to help me teach my Alg I students. It is great to provide them with two teachers' point of views. Thanks Prof Burger!

So now we have a sense of how to find zeros of polynomials and what they even mean. So zeros of polynomials are just the places where the polynomial crosses the x-axis, or those values for x, which if you plug into the polynomial give zero. So you can just take the polynomial, set it equal to zero, solve for x, and you’ve got it.
Okay, but let’s take a look at the possible answers that you can get when you actually do that solving. Let’s just think first about the quadratic and parabolas, since we’re sort of familiar with those. So if you take a look at a parabola like this, graphically, what you’d see is the roots of the parabola or the zeros of the quadratic would be those places where, in fact, the curve crosses x-axis, so you’d see them here, there are two. So it turns out there will always be two zeros to a quadratic. Now, those two zeros may, in fact, be imaginary numbers, in which case the picture would look more like this. It won’t actually cross the real x-axis, but there would be these two imaginary solutions out there somewhere. The other possibility, the more popular one, for at least me and probably for you, is that there are two real solutions, and there you go. You can see those two points.
Now, there’s a third possibility that’s sort of fun to think about, and that’s the one where the curve just touches, just grazes the x-axis. We saw this earlier when we were playing match game. Here’s another example of it. Now, in this case, believe it or not, we say--mathematicians don’t know how to count--we say that there are still two zeros, but they both happen to be the same. We say, “That’s a zero and that’s a zero.” So we just count that same zero twice. The reason why we count it twice is because you see the thing that sort of comes down, just nix it, and it goes up. Notice that if it were just a little bit lower we would have two, and so if you go up a little higher, we still have two, and so when you keep doing that, basically, mathematicians say, “There’s still two, but they happen to be equal.”
In this case we say that this is a zero with multiplicity 2. It just means that it’s a zero that actually happens twice. So this is also a zero right there that also happens twice. You can imagine with a cubic seeing the same kind of phenomena. It would look like this. You’d go up, you’d come down, and just caress the x-axis, and then go back up. This actually has three real roots, three real zeros. One of them is here, and then two of them are right there. This is a zero with multiplicity 2, because it comes down, just hits it, and comes up.
So, in fact, we can talk about the multiplicities of zeros, and that’s what I wanted to talk to you about just now. Let’s take a look at some examples. For example, let’s look at x² - 4x +4. And I want to know what are all the zeros of this object and what are the multiplicities. So I set this equal to zero and solve, and I hope, hope, hope that I can factor this, because if I can’t factor it this is going to be bad news. This plus sign tells me that we’re going to have both the same sign and they will both be minus, so I have a minus, a minus. Something whose product is 4 and sum is -4, so that would be 2, 2. And look what I see. Well, I’m going to write this out in this sort of compact way. I see that I have (x - 2)². So what are the two roots? What are the two zeros of this? Either x = 2 or x = 2. so this actually has a zero of x = 2, but it’s with multiplicity 2, because it happens twice, and that little 2 tells you that--it happens twice.
So here’s an example of a parabola that’s going to come down and just touch the world at that one point. The mathematicians would say it touches it sort of twice--once here, and then once at itself. How about a cubic? Let’s look at 7x3 + x. Let’s find all the roots and the multiplicities. So I would set this equal to zero and try to solve. Well, gee, factoring this looks a little bit tricky. It’s a cubic. How do you do that? I don't know. However, look, there’s a common factor of x. Let’s begin by just pulling that out. If I pull out the x, what I see is a 7x² + 1 = 0. And you may be saying, “Now, wait a minute. Where is the +1 coming from? Shouldn’t it just be zero if I take out the x?” No, because when I distribute this back I’ve got to have that x there. And so there’s an invisible 1 multiplying the x, and so when I pull out the x, don’t forget your special invisible friend, 1.
Now, what can I do with this? Well, I can factor this some more or I can just set everything equal to zero. This means x = 0, so x = 0. There’s one solution, and the other solution is this, which would mean that 7x² + 1 = 0. And we can try to factor that, or what I’m going to do is just bring the 1 over to the other side as a -1, and I would see 7x² = -1, divide both sides by 7, and I would see x² = -1/7 and then what happens? Well, I could just take ± square roots, but notice what’s going to happen when I do that. I’m going to see that x = ± the square root of -1/7, which is imaginary. And so what I see here is ± i so, in fact, I see that there are three different roots. The root x = 0, the root x = i, and the root x = - i. So these are three roots and they all appear just once, so I’d say each of these roots have multiplicity 1. They just appear once. There’s no coming down and just touching the x-axis. These cross right through, except these are imaginary, so in fact, we’re only going to cross the x-axis once, and these are going to be somewhere out in no man’s land.
One last example. This one’s real easy, so don’t worry about it. Suppose I just tell you that I have a function f(x) and I give it to you in factored form and I want you to find the zeros of this polynomial, but also the multiplicities. So there it is: (x + 1)²(x - 13)(x² - 10). To find the zeros I set that whole thing equal to zero. So I just set it equal to zero, and what happens? If I set it equal to zero, well either this term equals zero, or that term equals zero, or that term equals zero, because I have a product of numbers giving zero. Well, if this term equals zero, the only way that can happen is if x = -1. So x = -1 is a zero, but it has multiplicity 2. It actually occurs twice. If I were to write this out I could say (x + 1)(x + 1). So I see that the -1 appears twice. So I say this appears with multiplicity 2.
What about this? Well, here I would see that three times: x - 1, x - 1, x - 1. So the root or the zero, x = 1, that’s what makes it a zero, would actually have multiplicity 3. And what’s the solution here? The solution here is going to be what? Well, x = ± the square root of 10. So these both occur with multiplicity 1, because they only appear once. All right. Try some of these on your own and see if you can start to find zeros and multiplicities.

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