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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 athttp://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.
Okay, I thought I would just take second or two to show you some special cases of multiplying things out where in fact the answers are really sort of simple and so sometimes it's worth keeping your eye out for these things. So the first one let's take a look at is if you take a binomial and multiply it by itself. So just a square binomial, what would that be? Well, again, these are all just special cases of the more general thing.
For example, you could just foil that out, in fact that is how I would do it. So it is just (A + B) (A + B). And then you can foil or do it anyway you want, so in fact this is how I think about it I'd say, A x A = A^2. Now you know what I do, once you sort of start getting into it, those inside terms and the outside terms always combine together unless something really weird is going on here. So, for example, here I see, well, if I say it alphabetically, B x A, and then I also have a B x A out there, so if I have one B x A and then I add to it another B x A, I see 2B x A. Now some people might be tempted to say, oh, that's B x A or A x B squared. But that is not right, because remember we are adding here so this is an AB + AB = 2AB, and then B x B = +B^2. So in fact there's a neat thing whenever you have A + B, something plus something, and you square the whole thing what you get is this.
Let me show you the most famous of all classic mistakes, you are going to love this. And I really urge you to make this mistake right now with me. If you take (A + B)^2, of course what you get is A^2 + B^2, this is by far the number one most classic mistake that anyone could ever make, the squaring mistake, there it is. And that is just saying, (A + B)^2 = A^2 + B^2, just sort of distributing, if you will, that 2 across, the exponent 2. Not right, because you see that in fact you forgot about that middle term right there, this is absolutely not right. So always remember to foil, don't forget foil. This is a classic mistake so it is important to note that.
Now you could actually use this if you wanted to. Oh, look at that I ruined it, it was so nice, I had the whole thing there now it looks like A^2 + 2AB + a little teeny B, well, that stinks. In fact, I think, actually that is a philosophical point right there, why memorize that formula and have it there when you can just do the thing out live? So for example, let me just do a real fast one for you to show you that really it is better to just do it live. Suppose you want to multiply 2x + y by itself, what would you do? Well, you could memorize that thing, which was something squared, plus two times the something, times the something, plus the last thing squared, yeah right. Let's just write it out and either do your foil method or do the distributive thing whatever you like. These are really fast too watch how fast you can do it, 4x^2, the inside term produces a 2xy, the outside terms produce another 2xy. If I have 2xy and another 2xy that gives me a total of 4xy and y times y is y^2, but you can do it much faster than that, there's the answer there you are.
All right, now cubes, cubes, okay a little bit more tricky, let's take a look at taking a binomial and cubing it. Now this is not for the meek if you have a queasy stomach, if you were eating in the dorms recently you might want to stop, because this actually requires a teeny bit of work. Why? Well, because what you have to do is first square the thing and then multiply that three-term thing, that trinomial by another one of these. So in fact if you square that out we have the A^2 + 2AB + B^2, which by the way it's not that I memorized this it's just that I'm really doing it fast. And now if you distribute all the stuff, remember how you do this now, you have to take the A and it has to hit every single person, you have to do the B and you have to hit every single person, how many people should there be? There should be 6, because there are 2 people here, 3 people here, and everyone's going to have a little picnic, and so what you are going to have is a 6.
So let's see what we have going on here. What we have here is A x A^2 = A^3, then I have A times this, which is +2A^2B, then I have A times this, which is +AB^2, that takes care of A, A has hit everybody. Now let's bring in B, if I have B hit everybody I have B times this, now I'm going to write these things alphabetically just so that we can see what is going on here. So this is +A^2B, that is actually BA^2, but multiplication can go either way, here I am going to see +2AB^2, and the last term is +B^3, how many terms are there? There should be six, one, two, three, four, five, six monomials, looking good.
But notice, we can actually combine, it's like adding two polynomials, because these are all the same term. A^2B and here I see an AB^2, so if I add those up, oh boy, not much room, I would see A^3. And then if I have a 2A^2B and add another one, I would have 3. So, I would see +3A^2B and then if I add these two things I would see 3AB^2 + B^3. So in fact if you take A + B and you cube it, what you always get is an A^3 and a B^3, but then a lot of stuff in between, in particular, 3A^2B + 3AB^2. If you want to memorize that, you can. My advice is not to memorize that, but instead just to think about it.
In fact, if you were to now look at A - B, you can sort of do that out and you would see quite a different formula. All the terms would be the same, but the signs would be a little bit different, the signs would be a little teeny bit different. In fact I'll just illustrate that with an example and then we'll close because this stuff, you know, really, don't waste my time.
Here we go, how about the following, let's take 3x^2 - 2y and let's cube that whole thing. I'm going to see this as a two-step process. First I'm going to sort of take this and I'm going to multiply it by that thing just squared. Again, always notice that I am reducing the complexity, I just do something easy first and then bring on the harder thing. Now let's see how much of this I can actually do. Okay, so let me square this out, because something cubed is something squared times itself. So now I'm going to do some squaring right now live for you. Imagine, if you will, that I have the thing right next to it so that I can sort of do it like this. So this is going to be 3x^2 times itself, that would be 9x4 believe it or not. The inside term is going to be (-2y) (3x^2), so that `s a -6x^2y. But then when I do the outside term notice I have a (3x^2) (-2y) so I have another -6x^2y, so -6x^2y + -6x^2y is not zero, but it's -12x^2y. And the last times the last, a negative times a negative is a positive, so I see +2 x 2 = 4y^2. So, in fact, that is just foiling out this thing squared, but since it is this thing cubed I have to stick another factor on it. And now my job is to push all that stuff through and hope for the best. Let's see, so I'm going to take this 3x^2 and push it all the way through, so I'm going to see 9 x 3 = 27, x to the what power? Well, I have x^2 times x^4, same bottom so I add the exponents, when in doubt write it out. There are two of them here, there's four of them here and when I put them all together I have six of them. This times this term is going to be a minus. And now I have to multiply 12 x 3 and I get 36, and I have an x^4 and a y. And that last thing is going to be a +12x^2y^2, live math, it's sort of fun in a sick kind of way, I don't know, I'm enjoying it. Anyway, that just took care of the 3x^2.
Now let's do the -2y, it has to push through every single person here and let's see what happens. So what I see here is a -18x^4y that's this term, and here a negative times a negative is a positive and I have 12 x 2 = 24x^2y^2, because I have the y times y, and then that last term is going to be a -8y^3. Now you can combine all this stuff, let me combine it using the combining pen, if you combined it what you see is 27x^6, oh my goodness, now you have to combine all these things. That's going to be a negative something, now you have to actually add, now if you can add 36 + 18 then you are better than I am. This is going to be I think ends in a 4, and I think it is going to be a 54x^4y. And then if I combine these two things what do I get? Well, let's see, I would get, by the way, am I really happy with everything here? Let's see, give me one second here, because this is a plus that comes from here, and where did this thing come from, that thing came from way over here, and that's fine. Okay, no, I'm happy, I'm sorry. So let's see, all right, 12 + 24 = 36x^2y^2 - 8y^3, so you can se how those negative signs sort of distributed itself a little bit here. If you want a general formula for it you could certainly figure it out, but I don't think you should bother. There's the answer, notice; by the way, the answer has how many monomials, one, two, three, four. Maybe it is the wrong answer; four monomials shouldn't there be six. Well, there were six, one, two, three, four, five, six, but two pairs combined. So don't worry about that the red is correct and that's how you can actually cube out things in this one. See you.
Using Special Products Page [2 of 2]