- [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.