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## Factoring using structure

# Identifying quadratic patterns

CCSS.Math:

## Video transcript

- [Instructor] We're
told that we wanna factor the following expression and they ask 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 where I'll 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 can 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. 'Cause notice if you square this you're gonna get nine X 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 here is two 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 asking us
what pattern can we use to factor the expression? 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 two X 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, two times
U V would be equal to, well, you're not gonna have any Y in it so this is not going to be two 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'd 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 wanna factor the following expression. And this one essentially has two terms. We have a term here and
we have a term here. 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 six X squared. That works. And if this is equal to V squared, well, that means that V is
equal to Y plus three. 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 six X squared plus V plus Y plus three times U minus V. U is six X squared. Minus V is minus, we could write minus Y plus three or we could
distribute the negative sign. But either way this might
make it a little bit clearer what we just did. We used a 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
wanna 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 wanna 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 would 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.