Factoring polynomials with special product forms
We need to factor 49x squared minus 49y squared. Now here there's a pattern that you might already be familiar with. But just to make sure you are, let's think about what happens if we multiply a plus b-- where these are just two terms in a binomial-- times a minus b. If you multiply this out, you have a times a, which is a squared, plus a times negative b, which is negative ab-- that's a times negative b-- plus b times a, which is the same thing as ab. And then you have b times negative b, which is negative b squared. So when you do that, you have a negative ab and a positive ab, they cancel out. And you're just going to be left with an a squared minus a b squared. Now, this thing that we have here is exactly that pattern. 49x squared is a perfect square. 49y squared is a perfect square. We can rewrite it like that. We could rewrite this over here as 7x squared minus-- and I'll do it in blue-- minus 7y squared. And so you see it's a pattern. It's a squared minus b squared. So if you wanted to factor this-- if you would just use this pattern that we just derived-- you would say that this is the same thing as a, 7x plus b plus 7y times 7x minus b, minus 7y. And you'd be done. Now there's one alternate way that you could factor this and it'd be completely legitimate. You could start from the beginning and say, you know what? 49 is a common factor here, so let me just factor that out. So you could say it's equivalent to 49 times x squared minus y squared. And you say, oh, this fits the pattern of-- this is a squared minus b squared. So this will be x plus y times x minus y. So the whole thing would be 49 times x plus y times x minus y. And to see that this, right here, is the exact same thing as this right over here, you could just factor 7 out of both of these. You'd factor out a 7 out of that term, a factor 7 out of that term. And when you multiply them, you'd get the 49. So these are-- this or this-- these are both ways to factor this expression.