Multiplying and Dividing Rationals - Problem 3

This product is going to be a little bit tricky to find because I have 3 trinomials, that’s a little intimidating. All I’m going to do though is starting with factoring and I’m happy because all three of these trinomials have leading coefficients of one that makes factoring a lot easier.

So this first numerator I’m going to rewrite as m and m I need two numbers that multiply to 6 and add up to 5, can you think of what they are? A lot of students would say 6 and 1 but be careful 6 times 1 is 6 but they don’t add up to 5. What I really want to use is plus 2 and plus 3. There is that first guy factored I’m going to rewrite this first fraction all together. m plus 3, m plus 2 over m minus 3.

Okay let’s look at the next fraction. The numerator here actually I’m going to use a different color, the numerator here can be factored like this. I need numbers that multiply to -3 and add up to -2. Well that’s going to have to be -3 and +1 because they multiply to -3 and add up to -2. Here these guys are going to be m plus 1 and m plus 2. Let me rewrite the second fraction in its factored form. (m plus 1)(m minus 3), (m plus 1) and (m plus 2). Okay I’m going to erase a little bit here so I don’t run out of space.

Okay so now I have my first fraction factored I have my second fraction factored and I’m ready to cancel out any factors that are the same in top and bottom. It’s okay if the factors on the top here and bottom here I can cancel those guys out into two different fractions as long as they are being multiplied. If this were an addition or subtraction sign I wouldn’t be able to do that.

Okay so stuff that’s the same on top and bottom looks like let’s see, on top I have an m plus 2, so those guys can go, sound effects optional. On top I have an m plus 1 those guys can go and an m minus 3. So everything got cancelled out except for that little factor m plus 3, oh my gosh that’s pretty cool. That’s my final answer. That whole big mess, that whole big problem is equal to just m plus 3. That’s why I like simplifying it can be a drag sometimes but you start with really nasty looking products and sometimes you get a cute little answer like m plus 3. That’s why I think Math is kind of fun you go from ugly looking things that are really complex to things that are a lot more simple.