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Factoring two-variable quadratics: grouping

Sal factors 5rs+25r-3s-15 as (s+5)(5r-3). Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

We're asked to factor this expression by grouping. Now, they mention grouping, we're going to see what grouping is, but we're going to see very quickly that we have to do this thing called grouping because you can't just factor this expression. If you look at these, each of the terms, all but one of them is divisible by 5. So you can't just factor out a 5. Not all of them are divisible by either r or s. This is only divisible by r, that's only divisible by s, that's divisible by neither. So there is no common factor across all four of these terms. That's why we have to group them into groups where there are common factors, and then see if that simplifies the whole thing. And there is a little bit of an art to recognizing when you can factor by grouping, but they've set this problem up nicely for us. So if you look at these first two terms right here. You have a 5rs and a 25r. These two guys clearly have some common factors. They're both divisible by 5, they're both divisible by r. So if I just wanted to factor this one out, or if I wanted to rewrite it as a product of two expressions, how could I write it? Well, I could write it as a product of 5r times-- what's 5rs divided by 5r? Well, you still have an s left over, you just have an s there. Plus-- what's 25r divided by 5r? Well, 25 divided by 5 is 5, and r divided by r is just 1. So 25r over 5r is 5. So these first two terms can be factored into these two expressions. And then let's look at these second two terms. Well, they definitely have a common factor, you have a negative 3 or positive 3 common to both of these. Let's just go with the negative 3. And our goal is really to factor it into a negative 3 times, hopefully, something very similar to s plus 5. And you might already be seeing that it's going to factor into s plus it. 5. So let's factor out that negative 3. So these two terms you can rewrite as negative 3 times-- what's negative 3s divided by negative 3? Well, you're just going to have an s left over. And then what's negative 15 divided by negative 3? Well, that's just positive 5. And just like that, we've grouped them and we're able to factor each of those groups, and then something interesting might pop out at you. And one, you can always verify that you factored this properly by distributing each of these expressions. Distributing the 5r times s plus 5, and the negative 3 times s plus 5, you'll get exactly this. But something maybe jumped at out you just now. You have 5r times s plus 5. Then you have negative 3 times s plus 5. So now this expression, we have two terms instead of four, right, this is one term, this is another term. And they both have s plus 5 as a common factor, so we can now factor out s plus 5. So this whole thing can be rewritten as s plus 5 times 5r. Right? If you take 5r times s plus 5, and you factor out the s plus 5, you're just left with the 5r. And then similarly, if you take negative 3 times s plus 5, and you factor out the s plus 5, or divide by s plus 5, you just have a negative 3, just like that. And then we're done! We've factored this expression by grouping. It's s plus 5 times 5r minus 3. And you can verify it by multiplying it out. If you distribute the s plus 5 onto each of these terms, you'll get this expression up here, and then if you distribute the 5r over there you're going to get that expression. If you distribute the negative 3, you're going to get that expression. So this does simplify to that, so we have factored