Main content

## Get ready for Precalculus

### Course: Get ready for Precalculus > Unit 2

Lesson 5: Factoring using structure- Identifying quadratic patterns
- Identify quadratic patterns
- Factorization with substitution
- Factorization with substitution
- Factoring using the perfect square pattern
- Factoring using the difference of squares pattern
- Factor polynomials using structure

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Factorization with substitution

Unravel the mystery of algebraic expressions with factorization using substitution! This lesson explores how to simplify complex expressions by identifying patterns and substituting variables. By using U+V² and U+V x U-V structures, you can easily transform and factor expressions!

## Want to join the conversation?

- Are the variables
**U**and**V**always used for quadratic patterns?(10 votes)- U and V are just shortening the function. If you do programming, you use functions so you don't have to always type the long code over and over again; you will call the function. In here, they are just shortening each of the parenthesis into one letter.

Hope that helps!(3 votes)

- why isnt V equals to + or - 3y^3, Sal just put it as 3y^3(7 votes)
- Well, you are right. It CAN be 3y^3 or -3y^3. But it doesn't matter because when you are substituting either one into the equation (U+V)(U-V), the results are the same.

When Substituting 3y^3:

(U+3y^3)(U-3y^3)

When Substituting -3y^3:

(U+-3y^3)(U--3y^3) = (U-3y^3)(U+3y^3)

It's basically the same thing but switched the orders of (U+V) and (U-V). So the answer is that you are right, it's a good question though. It just doesn't matter.(4 votes)

- what exactly does it mean to "factor" something?(3 votes)
- It means to break something up into the product of two things. For example, if I wanted to factor 4, I could write it as 2 * 2. The same way, if you want to factor x^2 + 4x + 4, you could write it as (x+2)(x+2). Does this help?(7 votes)

- Does this make anybody elses brain hurt?(6 votes)
- math always makes my brain hurt ;-;(1 vote)

- Im now ahead of my math class a bit, so let me see if i understand this right.

- It's asking us to make a pattern

- (U+v)^2 is acting kinda like X?

-U falls into place for the first part of the problem and V for the last meaning 2u and 2v is the third part and we're seeing if it matches up?(4 votes) - I think using U and V, especially handwritten, can make it difficult to follow since they are so similar. Is this the standard variables for this sort of question or just a khan academy thing?(4 votes)
- Well variables are just symbols that represent a value that is still unknown. U and V can be any other letters. Whether or not it's standard, like "y=mx+b", I'm not sure.(1 vote)

- What do is really mean to factor something?(1 vote)
- To factor an expression means to break the expression into the product of numbers/functions that when multiplied together will equal the original expression.(5 votes)

- Why is y3 the square root of y6? 3•3 is 9, not 6.(1 vote)
- Because (y³)²=y⁶.

(y³)²=(y·y·y)²=(y·y·y)(y·y·y)=y⁶.

When two exponential expressions with the same base are multiplied, the exponents are added, not multiplied, and so taking the square root of the expression halves the exponent instead of taking its square root.(3 votes)

- Hello. At3:03in the video you said that the parentheses are not needed for "x+7" in the fully factored form of the expression. Wouldn't that change the entire expression? Isn't ((x+7)+y^2)^2 different from (x+7+y^2)^2? I was doing a problem very similar to this in a lesson and it counted me wrong because I had put parentheses around the x+c term.(0 votes)
- Hi, I don't think that it would be wrong because when doing ((x+7)+y^2)^2 you would use order of operations and calculate the y^2 first and you would also do that when doing (x+7+y^2)^2. then after you square the y you would add the x and 7 and get the same result whether you add the parentheses or not. Hope this was helpful. :D(4 votes)

- Our school uses (a+b)^2 but ok(1 vote)

## Video transcript

- [Instructor] 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. All right, 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 you'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 2UV plus V squared. If you've never seen this before or 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 U would have to be equal to X plus seven and when I say, actually I
should be a little careful. Can we express this entire
thing right over here as U squared? If U squared is equal
to X plus seven squared, that means that U is going
to be equal to X plus seven 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 could 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
two times U times V, 2UV? Well let's see. Two times U would be
two times X plus seven times V times Y squared and that's exactly what
we have right over here. It's 2Y squared times X plus seven. So this kind of hairy looking expression actually does fit this
pattern right over here. So you can view it as U plus V squared where U is equal to X plus seven and V is equal to Y squared. Now using that, we can now
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 seven, and I'll put it in
parentheses just so you see it very clearly, plus V
plus Y squared squared, 'cause 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 seven 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. 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 term, 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 U
would have to be equal to the square root of that
which would be two 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 9Y to the sixth. With the square root of nine is three and the square of Y to
the sixth is going to be Y to the third power. And then we could use that
to factor the expression, 'cause 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 our V, minus 3Y to the third. So there you have 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.