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Factoring Complicated Expressions - Problem 3
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Factoring an expression when we have negative powers on our variables; so the thing we want to do when we’re dealing with negative powers on our variables is to somehow make them positive and the way we do that is by factoring out the largest negative exponent that we have.

So when you say factoring out the smallest thing they have in common, when we’re dealing with the smallest variable and exponent and x that they have in common. When we’re dealing with negative numbers, the smallest thing they have in common is actually the most negative.

So looking at this x to -2, x to -3, x to -4, we want to factor out the most negative, x to the negative 4. So we then want to figure out what we’re left with. We have 6 and then we have a power of x. Remember when we multiplied bases we have to add exponents, so we want to figure out -4 plus what is equal to -2, that has to be 2 and you can always check to make sure you did it right, x to the -4 times x squared is x to -2.

With the same logic, we want to figure out x to -4 times what is equal to x to the negative third multiply basis add exponents -4 plus 1 is equal to -3, so that works and then this term isn’t going to have any xs, we’re just left with -2.

So by factoring out our smallest power in x, what we’ve actually done is made all of our exponents’ positive which means you can now factor this out. If you have negative exponents left inside, check your work something isn’t quite right. So you want everything in the inside to be positive.

So now we just have a standard trinomial which we have to factor. The x to -4 come down stays the same and now we have two binomials that we need to figure out what they are. So 1 and 2 are our only options here, so those have to go in as 1 and 2 and then our factors of 6 are either 1 and 6, or 2 and 3.

Let’s start with 1 and 6. There is no way for us to throw in a 1 and a 6 over here to get a 1. We’re either pairing 1 and 1 and 6 and 2 it’s going to be a big number or 6 and 1 and 1 and 2 again not going to quite work. So we have to be dealing with 2 and 3. In order to get -1, we want to pair the 3 with the 1 that will give us 3 and the 2 with 2 to give us 4 leaving us with a possibility of a 1. So this has to be a 3x and this has to be a x.

Last thing we want to figure out is the sign. I forgot my 2, I did forget my 2 didn’t I? That should be a 2x right there. Okay, so last just figure out our sign. Here we have a 3x, here we have a 4x. We want it to be negative which means our 4 needs to be negative. This turns into negative, it’s positive and we can always check our work to make sure we factor this out right. Here is our 6x², here is our -2, we have +3x and -4x which works out to be –x and -4 comes on for the ride.

So factoring a expression with negative exponents, factor out the smallest exponent so the biggest negative and that should leave you with something that you can factor on the inside.

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