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Multiplying and Dividing Rational Expressions - Problem 3

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!

Simplifying a rational expression with a lot going on. So what we have here are three fractions all with polynomials that are either being multiplied here or divided over there. And what we really want to do is develop a sort of methodical approach into how we simplify this. So what I’m going to do is just go down the row and first take out any common factors we have from any of these terms.

So most things we could be able to factorize here are by making the numbers smaller or the powers in x smaller it’s going to make our life significantly easier. So for this first expression what I see is I can factor out a x leaving us with 2x² plus 3x minus 2. For our second one we can factor out our x² leaving us with 2x minus 1, continuing down the row, nothing I can do with this guy we can take out a 3 from this one leaving us with an x minus 5. From this one we could take out a 3 from the numerator leaving us with x² plus 4x plus 4 and lastly we can take out a 5x leaving us with x minus 2.

So we just took out common factors to begin with and now what I’m going to do is combine a couple steps. So what I want to do is I want to rewrite this. First as I go through factoring each of these out and then also turning my division sign into a multiplication by flipping this last expression over.

So going through this what we end up with is x and then we want to end up with a positive 3x so that will be a 2x plus 2 and then a minus 1. You can always double check we end up with 4x minus x just 3x and the -2 in our single term. X², 2x minus 1, so we factored the first expression. Times we want to factor to this expression as well, so this is x plus 2, x minus 5, noting we can do with this statement nothing we can do with this statement. 3 x minus 5 and our last statement turning into multiplication this means we are going to flip this over so our 5x times x minus 2 is going to come into the numerator and our 3 times, this turns into what plus 2², ends up with a denominator.

Now I’ve drawn my parenthesis which is perfectly okay because we are actually multiplying three things together and remember if we are multiplying order doesn’t matter so we could either multiply these two out first and then by a third or we can just do these two out. The order doesn’t matter for multiplying do I will just drop my parenthesis all together.

So now what I’m looking for is just common factors that we can cancel out. So let’s see what I have, we have 2x minus 1, 2x minus 1 that can cancel. X plus 2 cancels with one of these x plus 2’s, this x plus 2 cancels with the other one, x minus 5, x minus 5. Anything else that can cancel we have an x² down here and an x and an x so those can all cancel and what I have left in the top then is just a 5x minus 2 and just a 3 and a 3 in the bottom. So leaving us with 5 (x minus 2) strike that a little bit clear for you, and then just a 3 and a 3 in the bottom so that’s over 9.

So a lot of work in order to do there’s a lot of factoring going on but basically all we did was simplified each thing and each term up individually, factored it out cancelled by factors and then we are just left with our simplified form.

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