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College Algebra: More Synthetic Division


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About this Lesson

  • Type: Video Tutorial
  • Length: 7:42
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 83 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: Polynomials: Synthetic Division (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.

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Okay, so let's see the synthetic division idea really in practice and see how you do it. Not hard to do once you remember the method. So let's consider the following: x5 - 61x3 + 5x - 1. I want us to divide that by x - 4. Since I'm dividing only by something of the form x plus a number or x minus a number, I can use synthetic division, otherwise, no synthetic division. Don't make a really good mistake about just taking a really complicated polynomial like x3 - 2x, if that were down here, and try somehow some synthetic division. It won't work. Only in this special case. Step 1. You write down this number but flip the sign, so instead of a -4 I'm going to write a +4. Then you draw this cute little box. Well, half a box. You can draw the whole box if you want. You can make it really fancy. You can make a little house on it, a little roof, a little chimney, a family, but let's just keep this part of the thing here. That's all I care about. Now, what I'm going to do is run through and just report the coefficients. That's all that matters, the coefficients of these people, remembering that if someone's not represented we have to put in as their vote, a zero.
So I see one x5, so I put a 1 here. I see no x4, so I put a zero. I see some x3 right there, so I put in a -61. I see no x2, so I put a zero there, and then I see a 5, and then I see a -1. So that's what I write. Now, let's see what happens. So the first step is to bring that one straight down. Then I take 4 times 1 and write that answer here. And then I add these numbers. If I add these up I get 4. If I multiply these I get 16, and then I add here. So if I add -61 and 16 I get -45. Now, what do I do? Well, then I take -45 and multiply it by 4 and I get -180. And then I add, so I have -180, and I multiply that by the 4 again, and I get -720, and then I add these things up and I get -715. Then I multiply that by 4, and I get -2860 and then I add the -1, so I get -2861. Well, now I'm done. I just have to know how to read this off. When I take this polynomial, which has x5 as the highest thing and I divide it by something with an x, I'm going to start with an x4 as the quotient. So these are the coefficients starting with x4. So, in fact, it's 1 times x4 + 4 times x3 - 45 times x2 and so on. So the answer would be x4+ 4x3 - 45x2 - 180x - 715, and this number here, -2861, that number is the remainder. So the remainder is that number there, and that's the quotient, and that's the remainder. I just used the synthetic division method.
Let's try another one. How about if I take 6x3 + 10x2 + 17 and I divide that by x + 3? I see the bottom is x plus a number or x minus a number, so I'm okay to use synthetic division. What do I do? I switch the sign here, draw my little box, and then I just copy the coefficients. So I see a 6, I see a 10. Notice, I see no x term, so I put a placeholder of zero, and then I see a 17. That's all. Copy the first 6 down, take 6 times -3 and get -18, and then I add and see -8. Then I see -3 times -8 and I see a +24, and I add here and I get 24. And then I take 24 and multiply it by -3, and I get -72. And then I add here, and what do you get? Well, you just have -72 + 17. So you get -55.
Okay, so that's your answer. Now, how do you decode it? Well, the way you decode it is this is going to be an x2 term since I'm taking an x3 and dividing it by an x, I have an x2, so I'm going to have a 6x2 - 8x + 24 and the remainder is -55. So there's a neat way of doing all that complicated long division just by using the synthetic division technique. One last thing to consider. How about this one?
So I have 2x5 - 3x2 + 7 all divided by x2 -3. So again, I use the synthetic division method. I switch this thing, it becomes a 3. I go down here, I make my 2. I see a zero for the x4. I see a zero for x3. I see a -3 for x2, I see a zero for x, and I see a 7, and so I start working. Bring down the 2, 6, add 6, 18, so forth, so on, so on, so on, so on, so on. Is this right? The answer is absolutely not, because notice that this bottom is not of the form x plus a number or x minus a number. This is x2 - 3. Synthetic division is not going to work there. So, in fact, even though you may think, "Oh, synthetic division." No, the only way to do this problem, the only way to do this long division, in fact, is to actually do the long division and to take the x2 - 3 and long divide it by 2x5 + 0x4 + 0x3 - 3x2 + 0x + 7, and then start long dividing it all out. That's the only way to do this. Synthetic division only works for x plus a number, x minus a number.
All right. Be careful of that. Enjoy these synthetic division problems. Once you start getting into them, not that bad, and you'll see that, in fact, when we have an x - 3 and we want to long divide, sometimes just finding the remainder will be a really, really important thing. So synthetic division is actually going to be useful to us, believe it or not. So practice these things, make sure you get it down, and good luck.
Polynomial and Rational Functions
Polynomials- Synthetic Division
More Synthetic Division Page [1 of 1]

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