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Beg Algebra: Adding and Subtracting Polynomials


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About this Lesson

  • Type: Video Tutorial
  • Length: 9:53
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 106 MB
  • Posted: 12/02/2008

This lesson is part of the following series:

Beginning Algebra Review (19 lessons, $37.62)
Int Algebra: Polynomial Basics (3 lessons, $4.95)

In this lesson, we learn how to add and subtract polynomials. A very important tip to keep in mind is to simplify the polynomial by combining like terms (terms with the same variable raised to the same power). When adding or subtracting strings of polynomials, you are simply combining the individual factors. Remember when subtracting that you need to distribute the subtraction sign to every factor within the parentheses.

Learn about multiplying polynomials in another lesson in the Beginner Algebra series:

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, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations.

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...


Recent Reviews

Polynomials have never been so easy!
~ Dustin3539

Professor Burger does an amazing job once again! He explains the concept of "combining like terms" extraordinarily well. The reason his teaching style is so successful is he explains "why" a student most perform an action, which makes the "how" make sense and therefore immensely easier to grasp. Not to mention he is funny and entertaining, which makes the lesson actually fun to learn. Professor Burger, where were you when I was in High School Algebra??? Great job!

Polynomials have never been so easy!
~ Dustin3539

Professor Burger does an amazing job once again! He explains the concept of "combining like terms" extraordinarily well. The reason his teaching style is so successful is he explains "why" a student most perform an action, which makes the "how" make sense and therefore immensely easier to grasp. Not to mention he is funny and entertaining, which makes the lesson actually fun to learn. Professor Burger, where were you when I was in High School Algebra??? Great job!

Page 1 of 2 © Thinkwell Corp.
A lot of times we’re going to actually take different polynomials and combine them together and we’re going to combine them together in a variety of ways. We can add them, subtract them, and multiply them together. You divide them; you’re not going to get a polynomial anymore. I just want to talk about how you add and subtract polynomials, how you put them together and combine them and the secret rule it comes down to, combining like terms. That’s the whole thing, just to combine like terms. If you remember that little mantra, you’re all set.
Let’s take a look at some examples. For example, 7x3 + 3x + 2x3 - 5 - 6x + 7. This is a big long polynomial and the question is how can we simplify that? Well, the way we do it is just combine like terms. Look for similarities. That’s all you’ve got to do. Look for similarities. Here you see, what? Here I see I’ve got a whole bunch of x3s. In fact, I have seven of them and if you scan the horizon you see that actually here’s some more x3 s. I have two more x3s. You can put those terms together and if I have 7 x3s and I add 2 more x3s that leaves me with 9 x3s. I’m combining these things and making things nice and tight. Now let’s see. Here I have a 3x and if I scan the horizon I see a - 6x so I can combine those terms because they’re just x’s. These are x3s but these are just x’s alone. How can I combine them? Well, I just combine them. I have 3x and I take away 6x. That leaves me with -3x and the way that I do it actually ? I’m going to tell you exactly how I do it in real life. I don’t do this inline thing. I circle the terms. This is exactly how I do it. You might not want to do it this way but it just tells me what I’ve done and what I haven’t done. For example, I’d circle this term and circle this with the negative sign and all and then when I combine those then that would produce a -3x and those terms can be put together, but you’ve got to be careful of that sign. It’s a -5 together with a +7, which combines to produce the net game of +2. There you can see I simplified this complicated polynomial by combining like terms. It’s really important.
Now, the reason why we combine like terms is because it helps us in adding and subtracting polynomials. Let me show you with an actual example. Here are two polynomials that I want to combine, 7y2 + 5 + y - 6y2. There’s one polynomial in terms of y’s and I want to combine that, I want to add that to this polynomial, 10y2 - 4y - 8y3 - 8. There’s this polynomial and I want to add it to that polynomial. How do I do it? It’s nothing more than combining like terms because think about it. If I just remove all the parentheses, it becomes one huge polynomial like this was and just a matter of combining all the like terms. If we do that, we combine the like terms, what do I see? I look for all the like terms. Let’s start with the highest power of y, which actually, if you scan the horizon, highest power of y happens way down here, y3 and I have -8 of them. There are no other y3 so that’s just it stand alone and I’m adding so I’m just going to start with a -8y3 and then I mark off that that term has been taken care of.
Now what about the y2 terms? Well, here I have a 7 y2’sthen I take away a 6 y2’s and then I add 10 y2’s. What’s the net gain? Well, I start with 7 of them. I take away 6. That leaves me with 1y2 then I add 10 more. That’s the gain of 11. So + 11y2 and I mark it down. I can see which terms are still left. What about y’s? I’ve got a y here and a y here. I’ve got y - 4y. That means I’m in the whole 3 y’s. That’s -3y and then the constant terms. Now it gets messy. You have to look really carefully. That’s still naked and that’s naked and if I combine those two naked terms I see -3. Naked terms come together and you see a -3. Here is the final reduced, simplified version of the sum of these two polynomials. That’s all you have there and I have the powers starting at 3 and then going all the way down to the constant term.
Now subtracting works the exact same way. Let’s try a simple example here. Suppose I have (3x2 - 7x) - (11x + 8). Same exact idea, except you have to remember to share the negativity. That negative sign has to hit everything in these parentheses. This is really important. Here’s a classic mistake. They pretend the parentheses are not there and they take -11x and then they add 8. That’s not what it says. Those parentheses mean you subtract this and subtract that. If you want, you could think about this as distributing that negative sign just as you always distribute with parentheses when you have something in front and that negative has to hit that 8 and make it a -8. That’s the only thing you have to be careful of when you’re subtracting polynomials. Now, I see 3x2 - 7x - 11x - 8. That’s so important. Now you combine like terms. I only have these x2 so 3x2 ,but then I have a -7x and a -11x. That’s -18x and then I have just -8 and there’s the answer.
Let’s try one more real quickie. Here’s a little more abstract one. Now I’m going to give you one that has a lot of variables and the trick is not to get freaked out by all these unknowns. We just combine like terms. So a3 - 3a2 b + 3ab2 - b3 and then from that I want to subtract away the following, a3 - 4a2 b + 2ab2 - 3b3. It’s a lot of terms there. Two things we have to remember. I’m subtracting so I’ve got to share the negativity. Each of these signs has to switch and then we’re combining like terms and notice these terms, although they may look the same, are not like. This has a2 times b
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and this is a times b2. Here I have a times a times b and here I have a times b times b. Those are different terms. They can’t be combined. We have to be really careful and let’s write all this out. To combine this I see a3 - -3a2 b + 3ab2 - b3 and now I’ve got to share the negativity. I see -a3. Minus a minus we know is a plus, + 4a2 b, minus a plus is a minus, 2ab2, minus a minus is a plus 3b3. Now I’ve got to go through and combine like terms. Here we go. a3 - a3, so these actually combine to give zero. They’re gone. - 3a2 b, this is a b2. That’s b3. That’s 4a2 b and no others. So these two can be combined. -3a2 b + 4 gives me a net gain of 1 so that’s + a2b. What about these terms here? This is 3ab2. These two terms can be combined. Be careful of that negative sign. That’s part of it. I have 3 of them, I take away 2. That leaves me with a one, + ab2 and then I’m left with - b3 and then + 3b3. They combine to give me + 2b3 and then once I’ve done that you’ll notice that everything has been taken care of and so this in fact is the answer to this thing minus that thing.
Don’t be freaked out when you see lots of different letters and different powers, just look for similarities and combine like terms.
See you soon.

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