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CCSS.Math:

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 plus 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 alright so let's go with the first part of it so they say they can factor the expression as u plus V squared so how do we see this expression in terms of U plus V squared well one way is to just remind ourselves what u plus v squared even is u plus V squared this is just going to be a the square of a binomial and we've seen this in many many other videos this is going to be u squared plus two times the product of these two terms so 2 u v plus V squared if you've never seen this before 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 squared well if this is U squared then you would have to be equal to X plus 7 and when I say actually let me be a little careful can we express this entire thing right over here as u squared if u squared is equal to X plus 7 squared that means that U is going to be equal to X plus 7 and then this right over here would have to be V squared if this is V squared then that means that V is equal to Y squared because Y squared squared is equal to Y to the fourth so V is equal to Y squared now they already told us that this this this can be factored as the expression u plus V squared but let's make sure that this actually works is this middle term right over here is this truly equal to 2 times u times V to U V well let's see 2 times u would be 2 times X plus 7 times V times y squared and that's exactly what we have right over here it's 2y squared times so X plus 7 so this kind of hairy looking expression actually does fit this pattern right over here so you can view it as au plus V squared where u is equal to X plus 7 and V is equal to Y squared now using that we can out actually factor the expression we can write this thing as being equal to u plus V squared and we know what U and V are so this is this whole expression is going to be equal to u which is X plus 7 and I'll put in parentheses just so you see it very clearly plus V plus y squared 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 plus 7 plus y squared squared 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 plus V times u minus 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 plus V times u minus V is equal to well if this is unfamiliar to you I encourage you to watch the videos on difference of squares but when you multiply this all out this is going to give you a difference of squares u squared minus V squared if you actually take the trouble of multiplying this out you're going to see that that middle term that middle third terms or the middle terms I should say cancel out so you're just left with the u squared minus the V squared and so does this fit this pattern well in order for this to be U squared and for this to be V squared that means U squared is equal to 4x squared so that means that you would have to be equal to the square root of that which would be 2 times X notice U squared would be 2x squared which is 4x squared and then V would have to be equal to the square root of 9 Y to the 6 with the square root of 9 is 3 and the square root of Y to the 6 is going to be Y to the third power 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 squared minus V squared so it's going to be equal to we can factor it out as or factor it as u minus or u plus V times u minus V so what's that going to be equal to so u plus V is going to be equal to 2x plus 3y to the third and then u minus V is going to be equal to 2x which is our u right over here minus RV minus 3y to the third so there you had it we factored the expression then you might want to write it down here but we just did it right up there and we're done