Fractions involving Quadratic and Linear Terms: Monic Expressions


It says simplify that fraction there and you can see we have a quadratic on the numerator.
Now before we get started guys, whenever we’re trying to simplify any type of algebraic fractions, always trying to factorize. So we just need to factorize the numerator and the denominator individually and then once we factorize it we can see if we can cancel anything out and make it more simple, okay?

So first thing I’m going to do is try and factorize the numerator. We can’t do anything more with denominator but we’ll do something with the numerator. So it’s a quadratic. So I’m going to use my usual cross method which you should be very good at by now. Some of you might not even need to use it. So I’m going to try and draw my two x’s and then you can see we always look at the last number don’t we the constant number, and it’s a positive, so it could be positive positive or negative negative but we’re not going to consider negative negative because this is a positive. So we don’t have to consider any negatives.

So positive positive, well 12 it could be 3 times 4, 2 times 6. I’m going to try and use 2 first of all, okay? So positive 2, so if I draw my cross, x times positive 2 is just 2x, isn’t it? So you just write that on the side, and 2 times 6 is 12. So the other one will be positive 6, and again I draw my other cross, so x times positive 6 is a positive 6x, okay?

Now can we make a positive 8x with these two? Yes, we can! 2x plus 6x is 8x. So we’ve got the right factors, so it’s going to be x plus 2, and x plus 6. So the numerator becomes x plus 2 times x plus 6, okay? So you’re trying to refresh your memory on those quadratic factorizations, okay? And then you can see guys, common, so all you need to do is just simply cross them out and we simply have x plus 6 left, okay? So after you factorize it, it becomes a lot more simple if you can just cross them out, okay? So that was question one.

Question two. Okay very very similar, it’s just a different expression. Again we have a quadratic on the numerator, let’s go ahead, and factorize that. So again, I put my two x’s, and this time it’s a negative, so it could be positive negative or negative positive. So 12, I’m gonna start with 4, positive 4, and I can see because you know 4 times 3 makes 12. So 4 is a factor.

So I’ll probably put 3 here. But let’s start with this. x times positive 4, we draw our cross, and we multiply them to get 4x. So definitely the other one will be negative 3 because positive negative makes the negative, isn’t it? So I put negative 3 there, draw my cross, and then x times negative 3 is negative 3x, and how do I make positive x with these guys? And you can see that 4 minus 3x is positive x which we’ve got there. So we’ve got the right factors.

So now it’s going to be x plus 4, x minus 3 on the numerator, okay? So because we just factorized it. And you can see that’s common and we just cross it out and therefore we just have x plus 4 left which is the answer, okay? Very easy!


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