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Multiplying and Dividing Rational Expressions - Problem 2
Whenever we are dividing rational expressions the main thing we want to do is basically flip our divisor and turn it into a multiplication problem. And after that all we need to do is try to factor everything out, cancel the factors we can because at simplify at the very end.
So I’m actually going to do two steps at once for this first one. What I’m going to do is I’m going to factor my y² minus 4 as I flip this as well. So what we end up is y minus 2, y plus 2 over 16 turns into a multiplication when we flip our fraction so then we end up with 8y over 2 minus y.
So now we just have to cancel any factors we have in common. Our 16 and our 8 cancel leaving us with a 2 in the denominator. And then what I see is that we have a y minus 2 and a 2 minus y. Remember these are opposites, I could factor in -1 out of one of them and it would turn directly into the other so when these cancel they actually cancel out to be a -1.
You could throw the -1 outside of either one it doesn’t matter because if a statement is negative the negative could be in the numerator denominator or just that up in front. So once we have simplified everything up as much as we can, we just have to multiply across see what we are left with.
So we have -y times y plus 2 in the top and then just a 2 in the denominator. If you wanted to you could probably distribute that -y through but for most purposes this would be perfectly fine.
So whenever we are dividing rational expressions, flip it over, turn into a multiplication, factor it out and sometimes you will have opposites where we have the same terms which switch your signs opposite, just remember those cancel to -1 and then just simplify through as much as you can.