Hi, and welcome. My name is Anthony Varela, and today we are going to be multiplying polynomials. So we're going to go through examples with distribution over several terms. We'll see an example of FOIL with some more complicated binomials that you may have seen before. And we'll also talk about multiplying three binomials, and this process could be extended to multiply more than three, if you wish.
So first we'll go over some examples with distribution. So here we have negative 2x, and it's being multiplied by this quantity, 3y plus 2x. So recall with the distribution, you multiply that outside factor by all of the terms inside the parentheses. So first, we multiply negative 2x by 3y. So we can look at those coefficients first. Negative 2 times 3 would be a negative 6, and then we have our x and our y, so negative 6xy.
Next, we multiply negative 2x by positive 2x. So once again, we can first look at those coefficients-- negative 2 and positive 2. That multiplies to negative 4. And then we have x squared, so minus 4x squared.
And now, taking a look at the proper order of how we'd like to put on our terms, remember we'd like to put them in descending order of their degree. Now both negative 6xy and negative 4x squared have degree 2, but we prefer to write that single variable term first. So I'm just going to switch these around. Negative 4x squared minus 6xy.
Now let's take a look at distribution over several terms. So we have three terms instead of two inside those parentheses. Well, our process is still multiplying that outside factor by everything inside the parentheses. So first, we multiply 2x by 3x squared. That's going to give us 6x cubed. Then we multiply 2x by 5x. That will give us 10x squared. And then we multiply 2x by negative 3, and that will give us a minus 6x. And taking a look at our terms, we have a degree three, degree two, degree one. So that's in proper order.
Now we're going to go through an example of FOIL. So here, we have two binomials that we're multiplying together. And our process for FOIL is multiply the first terms, so that would be these two terms, then the outside terms, and then the inside terms, and the last terms. And if you forget the steps of FOIL, you should think of FOIL as being distribution twice. So first we're going to distribute 3x into everything, and then we'll distribute 3 into everything.
So 3x times 2x squared gives us 6x cubed. Now we're going to distribute 3x to the negative 4. So 3x times negative 4 is negative 12x. Now we're going to distribute this 3 into 2x squared minus 4. So 3 times 2x squared will give us a 6x squared, and 3 times negative 4 gives us a minus 12. So now we're going to go ahead and put these in descending order of their degree. So we want to have degree 3, degree 2, degree 1, and then our constant term.
Lastly, I'd like to multiply three binomials. So here's our example, x minus 3 times x plus 2 times x minus 1. Now what we can do with these three binomials is recognize that we already know how to multiply two of them. So choose any two of your binomial-- obviously, it makes more sense if they're next to each other-- and FOIL them. And we can deal with that other third binomial later.
So let's go ahead and FOIL x minus 3 times x plus 2. So multiplying the first terms, we get x squared. Multiplying the outer terms, we get 2x. Multiplying the inside terms, negative 3x. And multiplying the last terms, negative 6. Now remember, this is all then multiplied by x minus 1.
So now this looks similar to our other example. But first what I'd like to do, actually, is combine some like terms. We have two x terms. So we have x squared minus x minus 6, and we're multiplying that by x minus 1. So what we're going to do, then, for that third binomial is distribute the first term and then distribute the second term.
So distributing this x into everything we see here, we get x cubed minus x squared minus 6x. And now we're going to distribute this negative 1 into everything we see here. So now we would have a minus x squared, a positive x, and a positive six. And notice one strategy is to align your like terms vertically, so we could add them quite easily. So combining these two, then, we have one x cubed, we have a negative 2x squared, we have a negative 5x, and then a constant term, plus 6.
So let's review multiplying polynomials. Well we talked about distribution, where you multiply out the outside factor by all of the terms inside the parentheses. We talked about FOIL as being distribution occurring twice. You multiply the first terms, then the outside, inside, and the last. That's what FOIL stands for.
We also looked into an example of multiplying three binomials. And you pick two of them because we already know how to multiply two binomials using FOIL, and then you can go ahead and distribute that third binomial by first distributing the first term and then distributing the second term. Thanks for watching this tutorial on multiplying polynomials. Hope to see you next time.