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College Algebra: Using the Binomial Theorem


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About this Lesson

  • Type: Video Tutorial
  • Length: 7:33
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 81 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra Review (30 lessons, $59.40)
College Algebra: Further Topics (12 lessons, $17.82)
College Algebra: The Binomial Theorem (2 lessons, $2.97)

To start with, Professor Burger will review binomials. Then, you will learn what the Binomial Theorem is and how and when to apply this formula used for writing out the powers of a binomial without manually multiplying it out. Additionally, you will learn what the relationship is between the Binomial Theorem and Pascal's triangle and how and when to use Pascal's triangle. To cement this knowledge, you will then work through an example problem by expanding (x+y)^4.

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 http://www.thinkwell.com/student/product/collegealgebra. 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

2174 lessons

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 www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


Recent Reviews

Funny, effective guy
~ Silvoir

Lol, the number one sound clip was worth the two bucks.

Anyways, good clip, doesn't waste time and doesn't go too fast.

Funny, effective guy
~ Silvoir

Lol, the number one sound clip was worth the two bucks.

Anyways, good clip, doesn't waste time and doesn't go too fast.

So, I wanted to return to this idea of actually sort of squaring out these binomials. You have a binomial and you sort of square it out or you might want to cube it out or something. So, if you just had a binomial, let me just remind you that a binomial, that’s a pretty easy thing. That’s just x + y. Binomial, bi; two terms, two nomials. Then I could ask you to square it and we did this a lot. You have to sort of use the foil business. Then we even took a look for a while at how you could actually cube this thing out. Which is a big deal, because remember (x + y)³ is not just x³ + y ³. Just like (x + y)² isn’t x² + y². In fact, this is my number one classic mistake. That’s right, number one on my top ten list is the squaring mistake.
So, (x + y )² doesn’t equal x² + y². You have those inside terms and here you’ve got a lot of inside terms. Well, you can see the progression here. I mean, what if you wanted to find x + y to the 110th power or 101st power? Well, now this becomes just a genuinely impossible problem because how could you multiply x + y out by itself 101 times? Well, you might be saying, “Gee, will I ever actually have to multiply x + y out by itself 101 times?” The reality is actually yes, in many, many situations you actually want to figure out what this thing equals when you raise it to a really high power. Perhaps you’re computing some sort of probabilities, you’re looking at some sort of function that you want to evaluate a certain point or expand or so forth and so on.
So, this is something that actually if you go on in quantitative endeavors you will have to face. Well, it turns out there’s this really cool thing called the binomial theorem that allows us to expand all these things out in a very simple way. I want to show it to you. In fact, I thought what I would do is just sort of inspire this by taking a look at what we have here. Let’s just take a look at the coefficients. So, for example, the coefficient here is just a 1 and a 1. Now, what would be the coefficients of this thing? Well, if I were to square that out what I would see is x² + 2xy + y². So, the coefficients would be 1, 2, 1. So, there’s that one. Now, what happens if you actually were to cube this out? Well, we talked about this awhile back briefly and it turns out you get x³ + 3x²y + 3xy² + y³.
So, there, and we worked that out and you can actually multiply that out and see that’s what you get, but there the coefficients would be 1, 3, 3, 1. So, if you write that down then I would see 1, 3, 3, 1. Actually, if you build this thing like a pyramid you see a really amazing pattern starting to emerge. Let’s take a look at it. Notice that what I always seem to be doing is always putting a 1 at the very, very beginning of the spot. So, I put a 1 here and I always enzyme with a 1. So, let’s make that just a rule. That in building this, what looks like a pyramid, let’s actually always start with a 1 and end with a 1.
Now, what do we put in between? Well, look what we did? We put a 2 right here in between the 1 and the 1. Notice that 1 + 1 = 2. Then I put 1, 2, 1. So, I put the ones at the end. I added these two things up and I got 2. What if I repeat that process? I put a 1 at the beginning and then I add these two things up, that gives me a 3. I add these two things up. That gives me a 3 and then I put the 1 at the end. Does that pattern continue? Suppose I actually were to continue this pattern. Well, I’d put down a 1 and then I’d add these two numbers up, I’d get a 4. I’d add these two numbers up, I’d get a 6. I’d add these two numbers up, I’d get a 4 and then I always end with a 1.
So, I’m building a triangle here. I’ll do one more. I’d put a 1 here. Then here I’d put a 5, because I’d add these two up. If I add these up I get a 10. If I add these up, I’d get a 10. If I add these up I’d get a 5 and then I’d put a 1 at the end. I’m building this triangle. This triangle is actually called Pascal’s triangle and it goes on forever but the really cool thing about Pascal’s triangle is it actually tells you the coefficients for when you’re expanding out a binomial to any power. So, for example, if you want to expand out a binomial to the third power you just go down three and those are the coefficients.
So, for example, if I say to you now, “Hey, what would this be, (x + y)4?” Well, instead of multiplying that out four times, which would be horrendous foiling and untangling and so forth. I can just go down to the fourth level here and those are the coefficients. All I’ve got to do is write out this polynomial in sort of a standard form. The standard form I would use is the following. This is always the x to the highest power term. So, this would be the x4 term. Then I have a plus and then this is the coefficient four. Now, I keep deducting one from the exponent on x and then increasing one of the exponent of y. So, watch what would happen next.
I would decrease the exponent here. So, I’d see x³, but now I’ll have a y to the first power. Plus, then I’d take this number. That tells me the coefficient 6 and then I would deduct one here. So, it’d be x² and increase this. This would be y². Notice that always if you add up the exponents you always get four. That should always happen. So, I’m just taking x4 and then x³ y and if you take the cubed and the first power and add them, you get four. If you take the 2 and the 2 and add them you get 4. That always will work out. Now, what’s the next term? Well, now I have a +4, I’ll continue here, + 4 and now what do I have? I have just x to the first power, but now I have a y³. Then finally, I have +1 times just y4.
So, look how easy I was able to actually expand this just using this Pascal’s triangle. It’s absolutely amazing. The binomial theorem is the theorem that actually says that these numbers are actually correct. So, if someone says, “What’s x + y to like the 10th power?” Instead of actually multiplying all that out and making the obvious mistakes that any human being would make in trying to compute all those variable running around, the better thing to do would be to start making this Pascal’s triangle and then keep going. It’d be a long, long thing, but it’s not that hard to do. You just keep adding these numbers and then very carefully start with x10 and then put plus that coefficient x9 times y plus the other coefficient x8 times y² and so on until you get down all the way to plus y10.
That’s all there is to it. So, in fact, expanding these binomials now are not that huge of a deal. Now, one thing that you may be wondering is how can you just find these numbers? Is there a formula for these numbers in order? Turns out there is and that really is the essences of the binomial theorem that will allow us to expand that out. I’ll tell you about that formula with binomial coefficients up next. I’ll see you there.

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