When I assign problems like this in my class, a lot of times students think that they are done in 10 seconds. They say oh! Okay cancel out 5x² and 5x² boom, so the answer is x plus 3 plus 10x. You guys that’s like the bonehead way to do this. What you need to do when you're multiplying rational expressions is multiply cross to top multiply cross the bottom and reduce. Also you want to make sure you are factoring everything.
So what I’m going to do is write this binomial as a sum on top of 1. That will help me see that what I’m really doing is multiplying two fractions. I have this fraction where I have x plus 3 on top, also on top of this guy I have 5x² plus 10x that’s all being multiplied, on the bottom I have 5x². So let me go through and try to factor everything that I can.
The top of the second fraction could be rewritten like this, 5x times x plus 2 because 5x is a common factor. So to rewrite this problem I’m going to have on top x plus 3 times 5x times x plus 2, all that stuff is in the top of my fractions on the bottom I have 5x². Okay my next job is to cancel out any factors that are the same in top and bottom. Well there is no x plus 3 terms so that guy is going to stay. This 5x term looks pretty close to that guy and then, I’ll come back to that, the x plus 2 term I can’t cancel anything so I know he’s going to stay. So I’m going to write him as part of my answer also.
Okay let’s go back to this 5x over 5x² you guys can probably do in your head that the fives are going to be eliminated, this x is going to cancel out one of those Xs and I’ll be left with just 1x in the denominator. That’s my most reduced form. Your teacher might want you to Foil this out and make that a trinomial it's totally up to your teacher or maybe the textbook instructions but this is the simplified version of this product.
Again the way students make errors in this kinds of problems is because they try to cancel out right away or because they don’t recognize that this could be written out as a fraction or because they factor incorrectly.
Be really careful with all those things guys especially if you have something where you have a rational expression multiplied by a polynomial that isn’t clearly a fraction yet. Just put it on top of 1 then it’s a fraction and you guys can do it.
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M.A. in Secondary Mathematics, Stanford University B.S., Stanford University
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