Factorization with substitution | Polynomial factorization | Algebra 2 | Khan Academy

We're told that we want to factor the following expression that they have right here, and they say that we can factor the expression as ( u + v ) squared, where ( u ) and ( v ) are either constant integers or single variable expressions. What are ( u ) and ( v )? And then they ask us to actually factor the expression. So pause this video and see if you can work on that.

All right, so let's go with the first part of it. So they say they can factor the expression as ( u + v ) squared. So how do we see this expression in terms of ( u + v ) squared? Well, one way is to just remind ourselves what ( u + v ) squared even is. ( u + v ) squared is just going to be the square of a binomial. We've seen this in many videos. This is going to be ( u^2 + 2 \times ) the product of these two terms, so ( 2uv + v^2 ). If you've never seen this before, I'm not sure where this came from. I encourage you to watch some of those early videos where we explain this out.

But does this match this pattern? Well, can we express this term as ( u^2 )? Well, if this is ( u^2 ), then ( u ) would have to be equal to ( x + 7 ). And when I say, actually let me be a little careful, can we express this entire thing right over here as ( u^2 )? If ( u^2 ) is equal to ( (x + 7)^2 ), that means that ( u ) is going to be equal to ( x + 7 ), and then this right over here would have to be ( v^2 ). If this is ( v^2 ), then that means that ( v ) is equal to ( y^2 ) because ( (y^2)^2 ) is equal to ( y^4 ). So ( v ) is equal to ( y^2 ).

Now they already told us that this can be factored as the expression ( u + v ) squared. Let's make sure that this actually works. Is this middle term right over here truly equal to ( 2 \times u \times v ), ( 2uv )? Well, let's see. ( 2 \times u ) would be ( 2 \times (x + 7) \times v \times (y^2) ). And that's exactly what we have right over here. It's ( 2y^2 \times x(x + 7) ). So this kind of hairy looking expression actually does fit this pattern right over here. So you can view it as ( u + v ) squared where ( u ) is equal to ( x + 7 ) and ( v ) is equal to ( y^2 ).

Now using that, we can now actually factor the expression. We can write this thing as being equal to ( u + v ) squared, and we know what ( u ) and ( v ) are. So this whole expression is going to be equal to ( u ), which is ( x + 7 ) (and I'll put in parentheses just so you see it very clearly), plus ( v ), plus ( y^2 ) squared. Because that's exactly what we wrote over there. And of course, you don't have to write these parentheses. You could rewrite this as ( (x + 7 + y^2)^2 ).

Let's do another example. So here, once again we are told that we want to factor the following expression, and they're saying that we can factor the expression as ( u + v ) times ( u - v ) where ( u ) and ( v ) are either constant integers or single variable expressions. So pause this video and try to figure out what ( u ) and ( v ) are, and then actually factor the expression.

All right, well, let's just remind ourselves in general what ( u + v ) times ( u - v ) is equal to. Well, if this is unfamiliar to you, I encourage you to watch the videos on the difference of squares. But when you multiply this all out, this is going to give you a difference of squares: ( u^2 - v^2 ). If you actually take the trouble of multiplying this out, you're going to see that that middle term — that middle third term or the middle terms, I should say — cancel out, so you're just left with ( u^2 - v^2 ).

And so does this fit this pattern? Well, in order for this to be ( u^2 ) and for this to be ( v^2 ), that means ( u^2 ) is equal to ( 4x^2 ). So that means that ( u ) would have to be equal to the square root of that which would be ( 2x ). Notice ( u^2 ) would be ( (2x)^2 ) which is ( 4x^2 ). And then ( v ) would have to be equal to the square root of ( 9y^6 ). The square root of ( 9 ) is ( 3 ) and the square root of ( y^6 ) is going to be ( y^3 ).

And then we could use that to factor the expression because we could say, "Hey, this right over here is the same thing as ( u^2 - v^2 )," so it's going to be equal to, we can factor it out as ( (u + v)(u - v) ). So what's that going to be equal to? So ( u + v ) is going to be equal to ( 2x + 3y^3 ), and then ( u - v ) is going to be equal to ( 2x - 3y^3 ).

So there you had it! We factored the expression. You might want to write it down here, but we just did it right up there, and we're done.