Multiplying and Dividing Rationals - Problem 4
I know this problem is going to be challenging because not only is it a division problem, so I have to flip my fraction here, but I have four trinomials that all need to be factored and some of them have leading coefficients that aren’t 1.
Holy cow my people this going to be tough but we can do it. Let’s go through it step by step. This first fraction I’m going to rewrite it because I don’t like how the denominator is not in standard from. The top is going to stay the same but on the bottom I’m going to rewrite it so that I’m starting out with t². By the way whenever I do problems that involve t, I write my Ts with a little curly q on the bottom so that it doesn’t look like a plus sign it's totally up to you but it helps me.
Okay so that’s my first fraction, all I did was rewrite the bottoms the way it was in standard form. Now dividing a fraction is the same thing as multiplying by the reciprocal that means this guy need to be turned upside down, what used to be on the bottom is now going to be in the top. 2t² plus 7t plus 3,
Okay my next job is going to be a factoring superstar. I got to factor all three of these or all four of those trinomials. Well this first one I noticed has a common factor of 5, so before I factor it with the two parentheses I’m going to factor out plus 5. T² plus 2t take away 3. Okay so now I can factor that numerator, it’s going to look like five times, let’s see my first two terms should be Ts. That’s a t, then I need numbers that multiply to -3 and add up to 2 like +3 and -1 that’s going to be my first numerator only. All I've done was factor that bit, don’t lose the 5.
On the bottom here I need numbers that multiply to positive 5 and add up to -6. Well since they add up to a negative value I know this is going to have to be two negative numbers like let’s see, -6 and -5, or no -5 and -1 that would work -5 and -1. Those have a product of +5 and they have a sum of -6.
Okay so that whole work everything that I have done so far has just been simplifying this first fraction. Now I got to simplify this second fraction doing a whole bunch of factoring. Okay so on top now I have this 4t² business and I have to figure out how to factor that. I’m going to guess my two first terms are going to be 2t and 2t but again with factoring there’s a lot of guessing and checking. I might be wrong that’s why it’s a good idea to use pencil.
Okay so now I need things that multiply to -5 and add up to -8 when I Foil. So this is where it gets kind of tricky I’m going to guess it’s going to be -5 and +1. I’m a pretty good factorer so I think that will work but I’m going to foil just in my head to double check. My first two terms multiply to 4t² my outers will be -10t, inners is +2 more Ts so that’s going to be -8 take away 5, good.
Okay last thing I’m going to have to do is factor the second denominator. Do you guys see how these problems are kind of a bear? Don’t worry you can do it my last term is 3, so I know these have to be 3 and 1 there we go. Now I have everything factored okay. Once everything is factored it gets to be kind of fun. Again this is Math land so it's kind of dorky fun. But I get to cross out anything that’s on the top of the fraction and in the bottom of the fraction if they are exactly the same.So if I had a 5 somewhere in the denominator I could cancel it out but I don’t. Those t minus 1s can go, let’s see here’s a t plus 3 and another t plus 3. Those t plus 1's can go and I think that’s it.
Okay so here comes my simplified final answer I think. On the top I have 5 and 2t take away 5 left, on the bottom I have t take away 5 and that’s it. A lot of students are going to be tempted to cancel something out like this t take away 5's. They are really tempting to cross out but be careful because t take away 5 on the bottom and t take away 5 on the top are not both factors. This is 2t take away 5, so that means like I can’t cancel anything out. This is my final answer this is the most simplified form of that whole original division problem.
This is a tricky problem you guys, it involves lots of different mathematical steps. If you get homework problems like this and you will make sure you guys mark which questions or which problems you're still stuck on, so when you go to class next time you can ask your teacher or you can ask your tutor. It always helps when you can bring in specific problems that you are stuck on and it will also help if you can bring in what part of those problems you are stuck on like maybe it’s the factoring.
The more information you can give your tutor or your teacher about where you're lost the more they’ll be able to help you be successful in your whole class.