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Factoring Complicated Expressions - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When asked to simplify expressions, sometimes we come across complicated expressions that are not easily factored by traditional methods. When factoring complex expressions, one strategy that we can use is substitution. When an expression has complex terms, we can substitute a single variable, factor and then re-substitute the original term for the variable once we have completely factored the expression.

Factoring complicated trinomials, okay? What I mean by complicated is something that sort of looks a little bit different than what we are used to. Okay?
The most common approach to doing a problem like this is at least that students want to do is to foil everything out, combine like terms and then factor it down again. Okay? But I want to show you a little bit of a shortcut that we can do in order to deal with this. Okay?
What we have is something squared minus something else times something plus 4, okay? And what we can actually do is make a substitution. And while I say I [IB] you as my substitution variable, you can do whatever you want but I highly recommend not using whatever variable is in your problem, okay? Because if you use x then you're going to get confused on what x is what and it'll get all confusing, okay. So if you just introduce a new variable for whatever you're dealing with.
Okay, by saying u is equal to 3x minus 2, I can then go back to this problem. 3x minus 2 squared just becomes u squared minus 4 times 3x minus 2 just becomes -4u and then the +4 is left down the end. Okay, we know how to factor this now, okay. So all we've done is we've taken something that's kind of complicated by making a substitution, a u substitution, I've turned it into something easy. I know how to factor this, this is just going to be u-2 quantity squared, okay? Be careful though because a lot of people want to end right here. They want to say, okay I factored it down, u-2. But if you look at it, our initial variable was x. It doesn't really make any sense to introduce a different variable as our end product, so what we have to do is go back and take this u and plug it back in, okay? So this u is 3x-2. So we end up with 3x-2-2. Combining like terms what we have then is 3x-4 quantity squared, okay? So factoring a fairly ugly thing by making a substitution makes your life easier. Like I said, you could foil all this out if you wanted to but it's going to be a lot harder and you're more likely to make mistakes than if you just make a simple substitution.
The other sort of complicated thing I want to look at is, if we're dealing with negative exponents, okay? So say I want to factor this expression which has 3 different negative exponents. What I want to do is draw a comparison. If I say 3x to the fourth minus 2x squared and I asked you to factor this. The first thing you want to do is to factor out the greatest common factor. You look for the smallest power of x that you have, okay?
So in this case you factor out the x squared leaving you with 3x squared minus 2, okay? You take out the smallest power on x. What we're going to do with this one is exactly the same except that it's sort of a weird thing to wrap your head around and that the smallest power of x is actually the most negative. [IB] think of a number line, your bigger numbers on one end, your big negative is on the other. The smaller numbers are towards those big negatives. So what you really have then is you're taking out your largest negative power. So in this case x the negative eighth. Take that out and then we are left with 3x. And then again, think about when we're multiplying bases we add our exponents.
So we want to figure out what goes here that when we multiply these things together we end up with a -6. -8 plus what is equal to -6. That should be a 2, okay? So then we have a +3 and again -8 times what is going to give us, alright x and -8 times what is going to give us x and -7. Again when we multiply bases we add exponents. -8+1 is -7 so this is just going to be x and then end up with -20. Okay?
One way you can always check to make sure you did this right is the whole goal of factoring out the negative exponent is that every other exponent is going to be positive, okay? So I factor out the -8, I'm left with the square, a single and a constant term. So I know that I did it right. If I still had negative exponents in here, something went wrong.
Now what we have is we took out this negative term and we have something we can factor, okay? For our purposes right now I'm not going to finish this problem. Hopefully you can see how to that but what I wanted to talk about is just how you tackle a problem like this. You factor out to the largest negative possible leaving you with something that you can then factor.

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