Strategy in factoring quadratics
- [Sal] In the last video we looked at three different examples, really as a bit of a review of some of our factoring techniques, and also to appreciate when we might wanna apply them. We saw in the first example that it was just a process of recognizing a common factor. Once we factored that out, we were done. In the second example, there was a common factor, four. Then, after that, we used, you could say, our most basic factoring technique, or one of our more basic factoring techniques, where we say, what two numbers add up to the first-degree coefficient, and then their product is the constant. And we were able to factor the expression. Then, in the third example, we, once again, started off by factoring out a common value, which, in this case, was three. We could've done it the same way we did the second one, or we could've immediately recognized that this is a perfect-square polynomial. But either way, we were able to factor the expression. Let's keep going to see if we can tackle some other types of polynomials that might require some other techniques. Let's say we have the expression seven x squared minus 63. Like always, pause this video, and see if you can factor that. I've intentionally designed all of these so that you can check whether there's a common factor across the terms, and here they're all divisible by seven. If you factor out a seven, you're gonna get seven times x-squared minus nine. You might immediately recognize this as a difference of squares. You have x squared minus this right over here is three squared. Minus three squared. If the term difference of squares or how to factor them is completely foreign to you, I encourage you to watch the videos on factoring difference of squares or do a search on Khan Academy for difference of squares. But you will see, when you have a difference of squares like this, it can be factored as seven, this is just a seven out front, and then this part right over here, get a different color, this part right over here can be written as x plus three times x minus three. It is x squared minus three squared. Now, one thing to appreciate. This really isn't a different technique than the one that we saw in the previous video. If we just focused on x squared minus nine, you could view this as x squared plus zero x minus nine. In that case, you'd say: "OK, what two numbers "get me a product of negative nine and add up to zero?" If I need to get a product of negative nine, that means that they must be different signs, a positive and a negative; otherwise, if they were the same sign, you'd get a positive here. So they're different signs. Nine only has three factors. One. You could either have one and nine. There's only two combinations here. You could either have one or nine, and three and three. And, if you make one negative or nine negative, that's not going to add up to zero. But if you make one of these threes negative, that does add up to zero. Say, OK, my two numbers are gonna be negative three and three. So it's gonna be x minus three times x plus three. Once again, I'm just focusing on what was inside the parentheses right over here. You'd put that seven out front if we were doing this exact same expression. But if you recognize it as a difference of squares, it might happen for you a little bit faster. Let's do one more example. Let's say that I have two x squared plus seven x plus three. In general, when my coefficient on the second-degree term here is not a one, I try to see is there a common factor here. But, seven isn't divisible by two, and neither is three. So I can't use the techniques that I used in the last few videos or even over here, where I say: "Oh, there's a common factor", and get a leading coefficient of one. If you see a situation like that, it's a clue that factoring by grouping might apply here. Factoring by grouping. On some level, everything that we've just done now you could view as special cases of factoring by grouping. Factoring by grouping, you say, OK, can I think of two numbers that add up to this coefficient. a plus b is equal to seven. a times b, instead of just saying it needs to equal to three, it actually needs to be equal to three times this, three times the leading coefficient, the coefficient on the x-squared term. It needs to be equal to three times two. If you think about it, we've always been doing that. The other examples we gave, the leading coefficient was a one. When you took the constant term, and multiplied it by a one, you were just saying a times b needs to be equal to that constant term. If you wanna talk about it more generally, it should be a times b should be the constant term times the leading coefficient. In the introduction to factoring by grouping we explained why that works. You should never just accept this as some magic formula. It makes sense for a very good mathematical reason. But once you accept that, then it's useful to be able to apply the technique. So can we think of two numbers that add up to seven and whose product is equal to six? They're going to have to be the same sign since this is a positive value. And, they're gonna be positive 'cause they're the same sign, and if they're adding up to a positive value, they're both gonna be positive. Well, let's see, one and six seems to work. One plus six is seven. One times six is six. In factoring by grouping we rewrite our expression where we break this up between the a and the b. So I can rewrite this as two x squared plus six x plus, I could write one x. Actually, lemme just do that. Plus one x plus three. As you can see, the seven x, different color, the seven x has been broken up into the six x and one x. That whole exercise I just did is to see how we can break up this first-degree term right over here. But then, what's useful about this, is now, we can do the reverse of the distributive property twice. So, for these first two terms, in a different color than I just use, these first two terms, you see a common factor. Two x squared and six x, they're both divisible by two x. So let's factor out a two x outta those first two terms. If you do that, two x squared divided by two x, you're just gonna have an x left over. Six x divided by two x, you're just going to have a three. And then you have plus. Over here, this is a special situation where, x plus three, there is no common factor between x and three, so we'll just rewrite that, x plus three. When I put a parentheses on it, which is equivalent to writing it without a parentheses, you might see something else. I can undistribute, or I can factor out, an x plus three. What happens if I do that? I'm gonna get an x plus three. And then I'm gonna have leftover in this term, if I factored out an x plus three, I'm just gonna have a two x left over. Two x. And then, this term, if I factor out an x plus three, I'm just gonna have a one left over. Plus one. Lemme do it in that same color. Having trouble switching colors today. Two x plus one. And we are done. So, as I said, these are all various techniques. On some level, factoring by grouping is sometimes viewed as the hardest one. But I'll say hard in parentheses because everything we did is just a variation, really, a special case, of factoring by grouping. As you can see, it's all about, two numbers that add up to that middle coefficient on the first-degree term when it's written in standard form. Their product is equal to the product of the constant and the leading coefficient. If you do that, you break it up, it works out quite nicely where you keep factoring out terms. This one, on some levels, was a little bit more subtle, 'cause you had to recognize that this x plus three has a one coefficient on there implicitly. One times x plus three is the same thing as x plus three. And then see that you can factor out an x plus three outta both of these terms, and then, once you do that, you're gonna be left with the two x plus one. But all of these, if you really feel comfortable with this arsenal of techniques, you're gonna do pretty well. Frankly, if none of these work, well, you might already be familiar with the quadratic formula or you might be soon to learn it, but that's when the quadratic formula might be effective if none of these techniques work.