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Int Algebra: Add-Subtract Rational Expressions


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:38
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 125 MB
  • Posted: 12/02/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Basics & Prerequisites (37 lessons, $52.47)
Intermediate Algebra Review (25 lessons, $49.50)
Intermediate Algebra: Operations with Rationals (3 lessons, $4.95)
College Algebra: Working with Rationals (3 lessons, $4.95)

In math, a rational expression or rational function is any function which can be written as the ratio of two polynomials. These are inevitably expressed as fractions. In this lesson, you will learn how to add and subtract rational functions. As with any fractions, to do this, you'll need to find a common denominator. In walking you through examples of this type of addition and subtraction, Professor Burger will highlight things to watch out for and shortcuts that can help you along the way.

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

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Operations with Rationals
Adding and Subtracting Rational Expressions Page [1 of 2]
So just like with regular good old-fashioned fractions, multiplying and dividing them is not a big deal. You just sort of put them together or flip first or whatever. Adding and subtracting rational expressions requires a little bit of extra work, because we have to still get a common denominator. So it's the same theme as adding regular fractions, but now the common denominators will probably have variables and letters in them. But again, we just need to take it easy and get a common denominator.
Let me illustrate this with a simple example, a basic theme. Suppose I have the rational expression 3 x 1+ and I want to add that to the rational expression x1? I just can’t add the tops and add the bottoms, just like I can’t do that with fractions. So what do I do? Well, I have to get a common denominator. Well now, how would I do that? Well, a temptation may be to say, “Well, I’ll just add top and bottom 3,” add 3 here and add 3 here. Well, that’s a good idea, except when you add 3 or add anything to the top and bottom of a fraction, you actually change the value of it. The only thing you're allowed to do is multiply by 1. If you want to add something, the only thing you can add, which won’t change the value, is zero. So you can add top and bottom to zero, but it’s not going to change much. But you can multiply through by 1. So actually what you have to do is take a look at this and, to get a common factor, I’m going to have to multiply top and bottom of this by this whole thing, and then multiply top and bottom of this by this whole thing.
So if I do that, top and bottom here I’ll multiply by x, and then the top and bottom here I’ll multiply by x + 3, because that’s the least common multiple, that’s the smallest thing that they have in common. So I multiply x + 3 on the top and I multiply it on the bottom. And I’m using my parentheses here. So here they are. Now you’ll notice the bottoms are the same, so I can just add the tops.
Now, there may be a temptation to distribute, and even if you do it correctly, x2 + 3x, let’s curb that temptation. Let’s keep denominators as factored as we can, because there might be some cancellations. There may not be, but, as a general rule, don’t waste time on a quiz or an exam or just in life in general, don’t waste time expanding out denominators, unless there’s some really compelling reason, and I don’t see that here today.
x + x = 2x, and then I just have a plus 3, this is added to technically invisible plus zero, so just plus 3, and all over the common bottom x(x + 3). Can I cancel the x + 3 with the x + 3 here and just have a 2 on top? Looks good, looks good. And if it looks good to you, that’s exactly why this is classic mistake number 3. Don’t cancel unless it’s a factor. You can only cancel factors. The 2 is not a factor of everything. If I just put parentheses around there – in fact, look at this. If I just sneak those parentheses around there somehow, then I could have canceled. But those parentheses are not there. It’s a big difference and so I can’t do it. That’s just the answer, there it is.
Okay, how about you try one? How about z x 1+, introducing now two letters, plus z -x 1? Combine those, first get a common denominator and let’s see how you do.
Okay, well to get a common denominator, I’m going to multiply the top and the bottom by this bottom, which is x – z, and top and bottom over here by this bottom, which is x + z, so I’d see ()(zxz x z -x ?+, and then I add to it when I get a common denominator here, I multiply top and bottom by x + z, so I’d have x + z over the same bottom.
So what does that equal? Well now, I have the same bottom, so I can actually combine the tops. x + x = 2x and I have a minus z, plus z. Actually they add to give zero. So, in fact, I’ve got zero, and on the bottom I just have all that good stuff there. Let’s make sure we record it correctly here, so that’s going to be (x + z)(x – z). So there’s the answer. Now, if you actually do feel compelled to multiply the bottom out, it doesn’t have an attractive bottom – oops, sorry. If you multiply out the bottom, you just see a difference of two perfect squares. And as attractive of a bottom as this is, I still think this is the way I would like to see the answer if it were me. Anyway, both of those are correct.
Operations with Rationals
Adding and Subtracting Rational Expressions Page [2 of 2]
Let’s try one last one. Let me try to really stick it to you here, 127tt112tt122+???+ . Do you think that’s enough? Nah! Let’s add on 16t12?. There you go! Give that a try and see if you can find the least common multiple and how you have to get everything to have a common bottom. See if you can combine, subtracting and then adding. Give it a shot. It’s going to take a little bit, but it’s really worth trying, and then I’ll do it for you and you can see how it goes. But give it a shot first.
Okay, well this isn’t going to be pretty. Let’s see if we can make a go at this. I’ve got to factor everything down here. That’s the first thing I want to do. So if I factor everything, I’m going to have a t and a t. I have opposite signs, so it doesn’t matter how I put them in, and I’ve got to multiply to give 12, but combine to give 1, so what would that be? Oh, 4 and 3! And where should I put the 4? I want the bigger thing to be positive, so I’ll put a 4 and 3 here. That looks like it works to me. Now I'm going to subtract off and I'm going to factor this bottom, the second bottom. So I’ve got t and t, the same sign, they're both negatives, multiply to give 12, combine to give 7, so I think 3 and 4 will work pretty well. And the last thing happily is the difference of two perfect squares, so I can say (t + 4)(t – 4).
Okay, well that’s how we’re looking, and now what is that least common multiple? Let’s see, I’m going to need one of these. In fact, let me just write down the least common multiple. So the least common multiple, the smallest thing that I can put together––I’ll need one of these. I’ll need one of these. See, here I already have that, so I don’t need anymore of them, I have enough. What about this? Well, this I don’t have yet. It looks like I do, but I don’t. Do I need any of these guys? No, I’ve already got it in my bag, so I’m fine. So, in fact, this is the least common multiple, this is the least thing that I can use as the denominator for this thing. This is actually going to require a two-page spread here.
All right, if I try to get the common bottom here, let’s see what the common bottom would require me to do. This is going to be the common bottom everywhere. So that’s the common bottom, what do I have to multiply top and bottom here to get this? Well, I already have that, I need this. So top and bottom here is going to be multiplied by t – 4. And then I’m going to see (t + 4)(t – 3)(t – 4). So there’s that bottom I’m trying to get everyone to have. So multiply top and bottom here by this. Now I’m going to subtract off, but what do I need here? Got this and got that, but don’t have this. So I'm going to multiply top and bottom here by t + 4, and now I’ve got a t + 4 factor on the bottom, plus all of the other factors we had before, which we had great fun with. And notice the common bottom when I have this factored, and then plus the last one. Well notice I’ve got those guys already in the picture, but I need this one, so top and bottom by t – 3 all over the common, the same one as this.
So that’s where we are. So now the bottoms are all the same and so I can combine the tops. So now what I can do here is the following: I can now just combine these things, but remember I’m subtracting here. Let’s not make classic mistake number 4, which is to not subtract everything. You’ve got to share the negativity, so I’ve got to subtract the t and subtract the 4. So it’s all over the common bottom of (t + 4)(t – 3)(t – 4). I'm not going to multiply that out, and you shouldn’t either. That saves a lot of work. But now, let’s combine things. I have a t, and then I have a minus t, so that drops out, but then I have a plus t. So this all gives me just a net gain of t. And then I see -4, a minus another 4. Remember, I've got to share that negativity. So minus a –4 is –8, and then another –3 is –8, -9, -10, -11, so that’s –11 on top. And so that’s the answer. Any cancellations? I don’t think so, and I would leave it just like this.
So that was sort of an extended big problem, but at least it gives you a chance to see how to find that least common multiple and then multiplying tops and bottoms by the appropriate things to get that all in shape.
Anyway, I’m ready for a shower, maybe you are, too. I think we’ve had enough of adding and subtracting rational expressions.

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