Beg Algebra: The FOIL Method

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THE FOIL METHOD
I want to tell you about a way of multiplying two very special types of polynomials [sic] and I want you to think about it as really like cooking. Maybe some of you actually do some barbequing. For example, you can wrap it up nicely, nice and tight and then what you can do is you can take your flame and you don’t want too hot of a flame. You just want just enough and you do this for about 45 minutes and then when you’re all done ? we’re going to fast forward. Now I already have one prepared and what you have here is really a foil. That’s exactly what’s going on here. I think about foil: first, outside, inside, and last. First, if you open it up and so there is the head. There’s the first and then you’ve got the outside. You’ve got the inside and then if you open it up, you’ve got the last and there you have your dinner and there you have the method of multiplying two binomials together. That is the method known as foiling.
Let me show you this first, outside, inside, last with examples. Now, suppose you’re looking at (3x + 1)(2x - 3). This little jingle of foil will tell me exactly how to factor it. Remember the strategy is clear. The strategy is I take this blop and multiply it by the 2x, take the blop and multiply by the - 3. Once I do that, I’ve got to take this 2x and distribute it through and take the - 3 and distribute it through. Now a way of doing all that at once is just recognizing that every term here has to be hit with every term here and if I do that in a systematic way, foil. F stands for first so I take the first times the first. I hit those two terms together. When I do that I would see (3x)(2x) and I’d see 6x2. Then I do the outside terms. I multiply this by this, the two outside terms. I did the first terms together, now I do the outside terms. That gives me a net gain of -9x because (3x)(-3) is -9x. Then I do the inside terms. I add 1(2x), which is just 2x and the last terms, which is -3(1), which is -3. Notice I have four terms, which is good because I’ve two here and two here. I should have a total of (2)(2) which is 4. I can simplify 6x2 - 7x - 3. There you have an easy way of multiplying these two binomials by themselves by just using the foil method.
Let me show you some more examples. Even if you’ve got lots of variables this still works as long as you just have two terms here and two terms here. Here are two binomials and now let’s multiply them together. First times the first, that’s going to 15x2. Then the outside terms, 5x(-5y). Be so careful here. That negative sign makes me a -25xy. My inside terms produces a -6xy and my last times the last, negative times a negative is a positive so I’ve got a +10y2. Again, four terms, perfect and I can combine the like terms and I see 15x2 and this gives me a net gain of -31xy and then + 10y2.
Then you can see how to actually multiply these people out using the foil method. Let’s try sort of a famous one. You’ll see this a lot in your life in various guises, not exactly like this. Let’s try this one. The foil method, here we go. The first times the first is x squared. Outside terms are + xy. Inside terms, notice the minus sign, is -xy and last times the last, negative times a positive is a negative, y2. This produced x2 but notice the inside and the outside terms cancel. They add to give zero. They drop out and I’m left with -y2. In fact, what I see here is the difference of two perfect squares. If you have x2 - y2, you can always factor it as (x + y)(x - y). You will use this a million times in this course. If you want to multiply two things that look like this and they’re the same except for a difference in sign, you know it’s going to have this general shape, the difference of squares.
How about one exotic example, (x2 + yz)(x2 - yz) but still there are only two terms here that I’m adding or subtracting. It should be a total of four and I can use the foil method to see exactly what they are. First times the first, that’s x4. Then the outside terms, -x2 yz. Inside terms, +x2 yz and then the last is a -y2 z2 and again notice that these cancel and I see x4 - y2 z2. Again, I see the difference of two perfect squares. This is x2 and this is yz all squared.
Again, the foil method is a really easy way of multiplying out these people, things of this sort when you’ve got two people here and two people here. Just do the first times the first, the outside times the outside, the inside times the inside, the last times the last. What you’re doing though in essence is just taking this blop, distributing it once, and then distributing these things back again.
I’ll see you at the next lecture.