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Int Algebra: Synthetic Division with Polynomials


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:00
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 86 MB
  • Posted: 12/02/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Intermediate Algebra Review (25 lessons, $49.50)
College Algebra: Polynomial & Rational Functions (23 lessons, $35.64)
College Algebra: Polynomials: Synthetic Division (2 lessons, $2.97)

You know how to use long division to divide two numbers; you can also use long division when dividing polynomials. In this lesson, Professor Burger will show you how to eliminate several of the steps in long division by u sing synthetic division. Synthetic division only works when you are dividing by (x + ?) or (x - ?) where ? is a number. In synthetic division, you start by using the coefficients of the polynomial in the numberator with switched signs. In the end, you will end up with the same answer for both the quotient and remainder as you would using long division, but it will be a less harrowing path to get there.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Intermediate Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers real numbers, equations and inequalities, exponents and polynomials, rational expressions, roots and radicals, relations and functions, the straight line, systems of equations, quadratic equations and quadratic inequalities, conic sections, inverse and exponential and logarithmic functions, and a variety of other AP algebra and advanced algebra.

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

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


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Exponents and Polynomials
Synthetic Division
Using Synthetic Division with Polynomials Page [1 of 2]
Okay, so in that last question of long division where I was dividing this thing into this long, long thing, what did we see? What we saw was that after doing all the long division we got this quotient and we got a remainder, and what I’d like to do now is actually work through that really long process, which is not by any means hard, but just long and complicated and you have to make sure you subtract everything, and see what really is important, and we’ll see that we can get rid of a lot of the steps. So let’s take a look down here and see what this looks like.
So here I’m dividing this thing into all of that. So I take it one at a time and then I subtract the entire answer, get down to here, and carry and so forth. What are the key things here? The only key thing is to know this number, to know the coefficients, there’s an invisible 1 here, a 0 here, a -4 here, a 7 here, and a -15 there. The fact that they’re x, x4, x3, x2, x and nothing, that’s really not that important as long as I promise to keep them all in order. And then notice when I subtract the only thing that matters is to take that 4 and multiply it by whatever the coefficient is here. So this coefficient’s important. That times that comes here, and then I subtract. And then I repeat the process. Whatever I have here I just take and keep subtracting and multiplying, and when I do that the important things, what I see, the real key people, this one out here, this -4, this 12, this -49, and finally, the remainder. Those are the important things, and we just got them by multiplying these things up here by this term and kept subtracting. For example, the 12 times 4 gave us 48, which we came here, which when we subtracted from this term gave us the -41. So each of these circled terms just came from a multiplication with one of these terms with 4 and then subtracting off one of these terms. So when we do that subtraction carefully, we actually see that all you need are these numbers and these numbers.
So I’m going to do the exact same problem, the exact same problem using a technique called synthetic division. With synthetic division we’re going to avoid all this long stuff and write everything basically on just two lines. So let me show you the analogous thing with synthetic division. So what I want to do is--I’ll write the problem up again, just to remind you. I have 415 -7x 4x - x24++x. Now, synthetic division only works when you’re dividing by something of the form x plus a number or x minus a number. That’s the only time it works.
Okay, now here’s the method of synthetic division. What we do is, first of all, write this number here, but write it with an opposite sign. So we always switch the sign of whatever’s there. So in this case I would write -4. And then I draw a box that looks like this, and here I’m going to actually write down all the coefficients that I see in front of these terms, but you have to promise me to put in the zero coefficients if the term is missing. For example, remember, there was no x3 here, so that means there’s a zero coefficient, a zero multiplying the x3, so I’ll write in a zero as a placeholder. So here I see a coefficient of 1, so I write a 1. Then I see no x3’s at all, so I put in a zero as a placeholder, like we did before. And then for an x2 I see a -4, so I write a -4. I see a 7, and I see a -15. Okay, so I just put the coefficients down in this fashion.
Let me recap what I did. The first thing I did was take this number here and write the opposite sign, -4, and then I copied down the coefficients, making sure that if someone wasn’t represented I put a zero in that placeholder. This is the x4’s placeholder, x3, x2, x, and then just the constant number placeholder. Here’s how synthetic division works, all I’m going to do is capture the multiplication and addition that we captured to get all this complicated stuff, and here’s how it works. Step 1 is just to bring that 1 straight down. So write 1. Now what you do is take -4 and multiply it by this and write the answer in here. So -4 times 1 is -4. And now all you do is add. If you add these two numbers you see -4.
Let me show you where we are, by the way, in the long division. See that -4 there? I’m going to show you exactly where that is. There’s the -4. I’m just trying to show you where this long, painful process I am. Right now I just found that -4. Now I repeat this process. I take this and multiply it by that, and in this case I would get 16--write it right here--and then add these two numbers, and so I get 12. Then I multiply these two things and I get a -48, and then I add these two things, and get a -41. Then I take -41 and multiply it by -4, and I get 164, and then I add these two things and get 149. And I claim that now we’re done. I haven’t told you what this means yet, but notice, this was a lot easier, I could fit it all on one sheet, than that other complicated thing. Here’s how you read this off. This is going to be the coefficient on an x3 term, because since I’m dividing a thing with x into a thing with x4, I know that my answer is going to begin with an x3. So, in fact, this is the coefficient on the x3, this is the coefficient on the x2, this is the
Exponents and Polynomials
Synthetic Division
Using Synthetic Division with Polynomials Page [2 of 2]
coefficient on the x, that’s the constant term, and this number is the remainder, and so the quotient would be x3, 1 times x3 - 4x2 + 12x - 41, with a remainder of 149. And notice, that’s exactly what we got here--look at that polynomial--and compare it to the actual answer we had, and there you see it 1x3 - 4x2 + 12x - 41 and the remainder of 149.
So, in fact, this really complicated, awful-looking thing can be simplified to this simple notation using synthetic division. Again, what’s the idea? First of all, it only works when you have x + something or x - something. If you had x2’s down here, x3’s down here, anything like that, it doesn’t work. But in this special case, all you do is take that number, flip the sign, write out the coefficients here and make sure you put zeros in place of placeholders, bring the first number down, multiply by this, write it here, then add, multiply, write, add, multiply, write, add, multiply, write, add, multiply, write, add. This term will be x1 less than this, and then you just keep going. So this will be x3 - 4x2, blah, blah, blah, remainder 149. This is synthetic division. We get the exact same answer, but it can fit on one page, and actually much easier to do than all that long division stuff.
All right, we’ll take a look at some applications of this and more of this type coming up next. Try it; see what you think.

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