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CCSS.Math:

we're told that we want to factor the following expression and they asked us which pattern can we use to factor the expression and U and V are either constant integers or single variable expressions so we'll do this one together and then we'll have a few more examples we're all encourage you to pause the video so when they're talking about patterns they're really saying hey can we say that some of these can generally form a pattern that matches what we have here and then we can use that pattern to factor it into one of these forms what do I mean by that well let's just imagine something like u plus V squared we've squared binomials in the past this is going to be equal to u squared plus two times the product of these terms so two u v and then plus V squared now when you look at this polynomial right over here it actually has this form if you look at it carefully how could it have this form well if we view U squared as nine X to the eighth then that means that U is and let me write it as a capital u U is equal to three X to the fourth because notice if you square this you're gonna get 9x to the eighth so this right over here is u squared and if we said that V squared is equal to Y squared so if this is capital v squared then that means that V is equal to Y and then this would have to be two times u V is it well see if I multiply u times V I get three X to the fourth Y and then two times that is indeed six X to the fourth Y so this right over a year is 2 u v so notice this polynomial this higher degree polynomial can be expressed in this pattern which means it can be factored this way so when they say which pattern can we use to factor this expression well I would use a pattern for u plus V squared so I would go with that choice right over there let's do a few more examples so here once again we're told the same thing we're given a different expression and they're asked they're asking us what pattern can we use to factor the so I have these three terms here it looks like maybe I could use I can see a perfect square here let's see if that works if this is U squared if this is U squared then that means that U is going to be equal to 2x to the third power and if this is V squared then that means that V is equal to five now is this equal to two times U V well let's see 2 times U V would be equal to well you're not gonna have any Y in it so this is not going to be 2 u V so this actually is not fitting the perfect square pattern so we could rule this out and both of these are perfect squares of some form one just has a I guess you say adding V the other one is subtracting V this right over here if I were to multiply this out this is going to be equal to this is a difference of squares and we've seen this before this is u squared minus V squared so you wouldn't have a three term polynomial like that so we could rule that one out so I would pick that we can't use any of the patterns let's do yet another example and I encourage you pause the video and see if you can work this one out on your own so the same idea they want to factor the following expression and this one essentially has two terms we have a term here and we have a term here they are both they both look like they are the square of something and we have a difference of squares so this is making me feel pretty good about this pattern but let's see if that works out remember u plus V times u minus V is equal to u squared minus V squared so if this is equal to u squared then that means that capital u is equal to 6 X 2 X 6 x squared that works and if this is equal to V squared well that means that V is equal to Y plus 3 so this is fitting this pattern right over here and they're just asking us to say what pattern can we use to factor the expression they're not asking us to actually factor it so we'll just pick this choice but once you identify the pattern it's actually pretty straightforward to factor it because if you say this is just going to factor into u plus V times u minus V well u plus V is going to be 6x squared + V + y + 3 times u minus V u is 6x squared minus V is minus we could write minus y + 3 or we could distribute the negative sign but either way and this might make it a little bit clearer what we just did we use the pattern to factor this higher degree polynomial which is essentially just a difference of squares let's do one last example so here once again we want to factor an expression which pattern can we use pause the video all right so we have two terms here so it looks like it might be a difference of squares if we set U is equal to seven and then this would be U U squared but then what can we Square to get 10 X to the third power remember we want to have integer exponents here and the square root of 10 X to the third power if I were to take the square root of 10 X to the third power it'd be something a little bit involved like the square root of 10 times x times the square root of x to the third power and I'm not going to get an integer exponent here so it doesn't look like I can express this as V squared so I would go with that we can't use any of the patterns and we're done