Beg Algebra: Multiplying Polynomials

This lesson has not been reviewed.

Please purchase the lesson to review.

Page 1 of 2 www.thinkwell.com © Thinkwell Corp.
MULTIPLYING POLYNOMIALS
Adding and subtracting polynomials, no big deal, all we have to do is combine like terms. Multiplying polynomials, it’s a whole different matter and I really want to show you now how to think about helping polynomials multiply. Let’s bring polynomials together and see what they can produce.
Now I want to warm up to this by looking at some very, very simple examples to get us warmed because maybe we are not world champions at looking at exponents. Let’s begin by looking at some exponents. Here’s two very simple polynomials. Let’s take the polynomial x2 and multiply it by the polynomial x3. Now what do you do in this case? What do you do with those exponents? Do you add? Do you multiply? Do you subtract? Who knows? When in doubt, write it out. All you have to do is write out what that means. What does that mean? There’s x and x, two x’s here and then I have x, x, x. What do you have? Well, together I have 5 x’s and you can immediately see that the rule is you add the exponents. Nothing to memorize here. If you get stuck just write out a simple example and you’ll see that when you’re multiplying x’s with exponents, you add the exponents.
What about this one? What if I have x4 and I raise that whole thing to the cubed power? Do I still add? What do I do? Again, when in doubt, write it out. This is x4, x4, x4. I’ve got three copies of them and what do you do with the exponents now? You know you add so in fact, here I’d see x to the, what? Well, 4 + 4 + 4, which is 12 and then I’d verify the fact that what we do in this case is multiply. Raise something with a power to another power you multiply those powers together. I’m taking x to a power and multiplying it by x to another power, I add the exponents.
Armed with that, we can now start to look at multiplying polynomials. Let’s take a look at a real simple thing. How about the polynomial x multiplied by the polynomial 2x - 3? What do you do here? Well, this operation is pretty straightforward and it really captures the idea what we’re going to do. I just take this and make sure it hits every single term here. I’ve got to distribute. I’ve got to use my distributive law. So x(2x) would be what? Well, I have x(x). That’s x2. I add the exponents. So I have 2x2 and then - 3x. There’s the answer and that’s all there is to multiplying any two polynomials. It’s as easy as that.
Let me show you with a more elaborate example and show you there’s nothing new, (3x + 1)(2x - 3). Now instead of the x in front times 2x - 3 I have a more complicated thing, (3x + 1)(2x - 3). How do I do this? The exact same way but now I think about that whole stuff there as a big blop. Now think about that like the x. What did I do? I distributed. I’m going to distribute this blop to this term and to that term. That’s all there is to it. Watch what I would see. I would see the blop times 2x and I’m going to write that out just like this. Watch. The blop times 2x and then - the blop times 3. The blop times the -3. Now, what is the blop again? You see it’s 3x + 1. That’s the blop. Look what I did. I just took this thing, whole thing with the parentheses and all, and multiplied it by the 2x and then multiplied it by the -3. It’s the exact same thing I did here with just the x alone, but now it’s more complicated so I cover it all up and I don’t think about it and just write it down just like I did before and then now there it is. Now I’ve got to do something else. I’m not quite done because I’ve got to now take these terms and distribute it back to remove those parentheses. There’s a second step in this process, namely I now take the 2x and multiply it by the 3x, take the 2x and multiply it by the 1 to remove these parentheses now. I distribute this way. If I distribute this way I’ll see a 6x2 + 2x. There’s that term and then this term. And then don’t forget that negative sign, - 9x and the negative minus 3(1) and now you can combine like terms if you want, which of course we always want to do. These terms can be combined together and I see 6x2 - 7x - 3 and that’s the product of these two polynomials. How did I do it? I just blocked everything together and distributed and then distributed back this way. That will always work.
Let me show you an even more elaborate one and show you that really it works, (2x + 3)(x2 - 5x + 4). I’ve got to make sure that every single one of these terms gets multiplied by every single one of these terms. In fact if you think of it that way, there are two terms here. There are three terms here. If every one of these gets hit with every one of these, there should be a total of six terms. When I’m done there should be six terms that I’m looking at and if not, I’ve made a mistake. Let’s see if we can do that. The way I’m going to do it is I’m going to blop this together and multiply the blop and distribute it right across. Distribute the blop to here, to here, and to here and when I do that, what do I get? I see the blop times x2 and then - 5x times the blop and then + 4 times the blop again. Now I’ll fill in what the blop was. The blop was 2x + 3 and now I repeat the same procedure except now I’m going to distribute these terms this way to get rid of those parentheses and when I do that, what I see is 2x(x2) = 2x3 and 3(x2) = +3x2. Now be careful of that negative sign, -5x(2x). That’s -10x2 and then -5x(3) = -15x. That’s where we are right now. Now over here, I distribute and I see a +8x
Page 2 of 2 www.thinkwell.com © Thinkwell Corp.
+ 12. How many terms do I have? I promised you six terms and I delivered it to you. Now we can just combine like terms if they’re out there. I only have one cube stuff so this is going to be 2x3. How many squares do I have? I’ve got this stuff and this stuff and that’s it. 3x2 - 10x2 is -7x2 and then I’ve got some x’s. -15 and then +8 so that’s -7 and then a + 12 and that is the product of this polynomial times that polynomial. It’s just a matter of taking one of them, thinking of it as one big blop, distributing it across, and then once it’s distributed, then take the individual terms and distribute across them and then combine like terms. That’s all that’s involved in helping polynomials multiply.
I’ll see you at the next lecture.