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Math is cool and you can do it.
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Okay, now we're going to be multiplying rational expressions.
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Just finish simplifying them, which was just like reducing fractions.
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This is exactly the same as like multiplying fractions.
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But actually, it's easier to always factor and cancel before you multiply fractions.
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And it's going to be the same thing here.
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So we have two fractions here. What I want you to do always is put parentheses around each name, rate, and denominator.
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Remember, you can only cancel the 8 minus x minus 12 with another 8 x minus 12.
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Don't try to cancel the 8 in the 14 or the 12 in the 7 or the anything else here.
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It has to be the whole factor.
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So what we're going to do, this is fun.
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You just get to do a bunch of factoring and then what's cool is a lot of it's usually going to cancel.
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So out of the numerators, I'm just going to write one big fraction.
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I'm just going to write the factorization of both numerators in the top and the factorization of both denominators in the bottom.
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All of these are only two terms.
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So we're going to look for common factors to see if that works.
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So let's see, out of the top, I could pick out a 4 times 2x minus 3 out of the first one.
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And out of 42x plus 21, I could take out a 21.
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And that would give me a 2x plus 1.
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All right, so I've done the numerator.
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And if you want, you could also write 4 times instead of just putting it in front.
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All right, now let's do the denominator.
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14x plus 7.
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You could take a 7 out and I have a 2x plus 1.
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Wouldn't you know what?
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There's going to be a common factor.
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And out of 32x minus 48, the largest factor out of that is 16 times 2x minus 3.
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Did you kind of guess that maybe it was going to be this 2x minus 3?
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You kind of think like someone writing a math book.
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Yeah, that's what's going to happen.
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Okay, so now let's cancel what we can.
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I could cancel this whole factor of 2x minus 3 with that whole factor.
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I can cancel this 2x plus 1 with that 2x plus 1.
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And we can also do some fact, the canceling with the numbers.
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The 4 and 16 here, for instance.
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I could cancel that and that gives you a 4.
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And with the 7 and 21, I can cancel that.
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I get a 3.
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And I didn't show it here, but with these, they just cancel until 1's right.
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That cancel's until 1.
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So gosh, the only thing is this kind of messy.
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Very carefully, it was left.
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There's a 3 and a 4, the rest are 1's.
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So my answer is going to be 3, 4's.
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So basically this original problem was a big messy way of just writing the number 3, 4's.
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Interesting way of thinking about it, don't you think?
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Okay, I want you to try this on your own, put it on pause.
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And you're going to take all four of these, put parentheses around them,
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get out your scratch paper, factor all four of them.
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These are the easier ones to factor at least.
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So when you do that, you should end up with four factors in the top,
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four factors at the bottom.
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See if anything cancels and see what you get for your answer.
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And then come back and see what you get.
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Okay, here are your factors.
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This is going to be x plus 7 times x minus 3.
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That's how you get x squared plus 4x minus 21.
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This will be x plus 5 and x minus 4.
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That gives you x squared plus x minus 20.
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Let's see the bottom one, let's see, factors of 28,
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have a difference of 3, 7 and 4.
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So it's going to be x plus 7 and x minus 4.
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And the last one, let's see, we're going to have to get x plus 5 and x minus 3.
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So those are the four factors you should have gotten.
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I just love this problem, it's so much fun.
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Look at the x plus 7's cancel, x minus 3's cancel,
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the x plus 5's cancel, and the x minus 4's cancel,
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everything cancels, what happens if it all cancels?
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The answer is 1.
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Hey, so this is a really bizarre way of writing the number 1.
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That's just a more complicated way of doing it.
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Okay, here's another one for you to try.
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So try it on your own first, remember to put the parentheses around all four of the numerators and denominators.
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Then yeast factoring methods, you're going to be using the difference of two squares in here.
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You're going to do pulling out the greatest common factor.
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You're going to have to factor a trinomial and two by no-mails.
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So you're really getting good at your factoring skills, I hope.
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And then see what can cancel from that.
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Okay, so what do we get here?
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For x minus y squared is x plus y times x minus y.
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And over here, two terms take out the greatest common factor, it's 3x.
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And what will be left is x plus 2.
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Okay, let's do the bottom now.
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Two terms, so let's try taking out the greatest common factor.
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That'll be 3x.
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And that leaves you with x plus y.
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All right, and then here we have a trinomial.
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It's one isn't so bad though because notice the last term is just a minus y squared.
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So you know it's going to be a y and a y.
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So we know we're having a 3x and a 3x and a y and a y.
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And because that's the only possibility for 3x squared.
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So we just have to figure out where to put the minus and plus sign.
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We want to get a minus 2x y, so it's going to be a minus and a plus.
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Is that it?
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I'll give you a negative 3x y and a plus y and x y.
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Yep.
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Okay, so those are the factors you should have gotten.
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And now we go across the top, see if we can cancel anything.
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So x plus y cancels with that.
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x minus y cancels with the other x minus y.
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Both of us, we have a 3x and a 3x.
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It could have been just maybe the 3s cancels or just the x's.
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But actually the entire thing cancels here.
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3x cancels with a 3x.
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And that's it.
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x plus 2 over 3x plus y is our answer.
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Resist the temptation to try to cancel these 3x squares right here in the first problem.
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That's where people make them mistakes.
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They want to just start canceling before factoring.
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You must factor before you can cancel anything.
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Factor first.
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Okay, here's another one I want you to try all on your own and then come back and compare your answer with how I do it.
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Okay, put it on pause.
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Go for it.
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You back?
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All right, so let's do.
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The numerator.
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9x plus 18, we could take out a 9, that's x plus 2.
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All right, now here we're going to have to factor.
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Now this is where you could do it over on scratch paper someplace.
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I'm going to just try.
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Let's see.
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I'm going to try 2x and 2x.
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Is that going to work?
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I do not think that's going to work.
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The reason it's not because if I put a 6 and 1 or a 3 and a 2, there will be a 2 in the same parenthesis.
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So I know that's not it.
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I'm doing it by 12 factors but you could do it by any of the methods for factoring.
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So I know it's going to be a 4x and x.
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And let's see.
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I think it's going to be 3 and 2.
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Does that work?
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Yes.
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All right, again, I've just factored a lot so I'm fairly quick at doing it by 12 factors.
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But you can go to your scratch paper and figure it out any way you want.
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So let's see.
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And at the bottom we take out an x to get 4x minus 3 and for x squared minus 4 it's a difference of 2 squares.
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So the rest of them aren't too bad.
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All right, so what do we do here?
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All right, there's a 9 and is there any factor down here I can cancel with a 9?
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No, don't try to cancel with this 3.
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That's not a factor.
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4x minus 3 is the factor.
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Is there an x plus 2 in the denominator?
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Yes.
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I can cancel.
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Is there a 4x minus 3 in the denominator?
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Yes.
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I can cancel.
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It's an x minus 2 in the denominator.
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Yes.
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I can cancel.
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What's left?
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9 over x.
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So our answer is 9 over x.
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So that is how we multiply rational expressions.
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We factor and then we see if we can cancel anything.
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So don't just start off trying to multiply.
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9x plus 18 times 4x squared minus 11x plus 6.
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Yikes!
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Way, way too hard and then you would get stuck in that in a how to factor it.
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Okay, that's it for this video.
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I hope you learned something.
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Remember how this school...