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Simplifying Complex Fractions - Part 2

In our previous video of simplifying complex fractions, we noticed that we had one fraction in the denominator and one fraction in the numerator. But notice, in this scenario, we do have one fraction in the numerator but we do have a fraction in the denominator, but then notice there’s this random x there. So what we need to is we actually need to get this represented as one fraction. Let's go over here, on the side and do just that. We have x over 1 minus 4 over x. All that we are going to do is find the least common denominator, which is x. Notice, this first fraction here doesn't have a denominator of x, so we must make it have a denominator of x. So I am just going to multiply by 1 in a really fancy way, by saying x divided by x. Multiplying across the top we have x squared. We don't have to do anything with this fraction considering we already have a denominator of x. So we can just bring down the negative 4. Now that this is written as one fraction, we can go ahead and apply it to the problem at hand. I am just copying down my numerator x plus 2, divided by x, and now I’m substituting in the x squared minus 4 over x. Now I can apply what the previous video said. Because now all I have to do is use the whole idea of "keep, change, flip" considering now that this is one fraction and this is one fraction. So, I’m going to keep the first, change the sign from division to multiplication, and then flip the last. So, notice, from here it just turns out to be a multiplication problem, and notice we have an x in the numerator, and an x in the denominator. So those’ll cancel out. So, what we’re left with is x plus 2 divided by x squared minus 4. Now, this is not its most simplified form. So in order to simplify it, we must remember that we have to factor as much as possible in these problems. So x plus 2 is already prime. However, though, the x squared minus 4 is called the difference of squares. So we say that x plus 2, divided by – and then factoring that, we have x plus 2 times x minus 2. So, now, once again, we see x plus 2 in the numerator and an x plus 2 in the denominator. So, we can cancel out since those are factors. So, remember, you can cancel factors, not sums. So, that’s just going to equal all that’s left over in the numerator, which is a 1, and then divided by what’s left in the denominator, which is an x minus 2.

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