Hi, and welcome. My name is Anthony Varela. And today we're going to multiply and divide rational expressions.
So we'll start by reviewing how to multiply and divide numeric fractions, so no variables to start. Dividing fractions is also the same as multiplying by a reciprocal.
So we'll talk about that. And then we'll apply all of these principles then to algebraic fractions. So we'll start by multiplying numeric fractions. And in multiplication, it's pretty straightforward. We just multiply across the numerators, and then multiply across the denominators.
So if you'd like to multiply 2/3 by 5/8, we'll multiply across the numerator to get 10. Multiply across the denominator to get 24.
Now our last step then would be to simplify this fraction. So we're going to identify common factors in both the top and the bottom, and then cancel them out. So I see that these are both even numbers. So they share a factor of 2. I can write 10 as 2 times 5, and 24 as 2 times 12.
Canceling out the common factor of 2, I get 5/12. And now 5 and 12 don't share any other common factors. So this is in its simplest form.
Now when we're dividing fractions, we write this as a multiplication problem. So 3/4 divided by 1/2. We're going to multiply the first fraction by the second fraction's reciprocal.
Now in a reciprocal, we just make the numerator the denominator, and make the denominator the numerator. So we're flipping it around. So to write this as multiplication, I would write 3/4 times 2/1. That's the reciprocal of 1/2.
Now we can just follow our process for multiplication. Multiplying across the numerator, 3 times 2 is 6. Multiply across the denominator, 4 times 1 is 4.
And we simplify by canceling out the common factors. They both share a factor of 2. So 6 over 2 is 3. And 4 over 2 is 2.
So 3/2 is 3/4 divided by 1/2. So now let's get into multiplying and dividing algebraic fractions. So we're going to follow our same processes here for multiplication and division of fractions. We're just going to have variables.
So our first problem is multiplication. And we'd like to multiply x squared minus 1 over x plus 3, times x minus 2 over x plus 1.
Now before I actually get to multiplying across our numerators and our denominators, what I find really helpful is if, at all possible, factor all of your numerators and your denominators. That's going to help you identify common factors in order to simplify later. So these are already written out in their factors.
But x squared minus 1 can be written as x plus 1 times x minus 1. So everything else has stayed the same so far. I'm just writing x plus 1 times x minus 1 instead of x squared minus 1.
So now we're going to multiply across our numerators. So I have x plus 1 times x minus 1 times x minus 2. That's what we have here. And multiply across our denominators, x plus 3 times x plus 1.
So now, we want to identify common factors and cancel them out. We see that both the top and the bottom include an x plus 1.
So I can cancel those out. And I have x minus 1 times x minus 2 over x plus 3. So just rewriting the fraction, but without the common factors here.
So I could leave it like this, or I can go ahead and rewrite this back in the standard form. So FOILing the numerator, I would get x squared minus 3x plus 2. And I have x plus 3 as my denominator.
So now let's go through a final example where we're dividing algebraic fractions. So we would like to divide 2x minus 4 over 2x minus 10 by x squared minus 4 over x minus 5.
So remember, dividing two fractions can be thought of as multiplying the first by the reciprocal of the second. So right away I'm going to write this as multiplication.
So my first fraction hasn't changed. But I've written the reciprocal of the second fraction and changed from division to multiplication.
So once again, before we multiply across the numerators and the denominators, I would like to factor as much as I can. So I'm going to factor out a 2 of 2x minus 4. So I have 2 times x minus 2. And then I'll tack on my x minus 5 when I multiply.
Now for my denominators, I'm going to factor out a 2 here as well, because 2x and negative 10 share a factor of 2. So that would be 2 times x minus 5 when I factor out the 2. And how can I factor x squared minus 4? Well, I recognize this as a difference of squares. So I can write this as x plus 2 times x minus 2.
So here I have not only combined or multiplied across my numerators and my denominators, but I factored out as much as I can. And now I'm going to simplify by identifying common factors and canceling them out. And this is interesting. There's actually a lot of common factors that we can cancel.
First, to start, I see a 2 on the top and the bottom. So I can cancel those out. I see an x minus 2 on the top and the bottom, so that's going to cancel out as well. And I see an x minus 5 on the top and the bottom.
So my entire numerator has been canceled out. That doesn't mean it's 0. There is still a factor of 1 that's implied. So my numerator simplifies to 1.
And then my denominator, I just have the factor x plus 2. So this would be my simplified quotient for this division problem.
So let's review multiplying and dividing rational expressions. Well, with multiplication, we can multiply across the numerators, multiply across the denominators, and then simplify our fraction by identifying common factors and canceling them out.
So you might decide to factor out before you multiply across. So you can easily identify those common factors.
If you're dividing, you're going to create your own multiplication problems. So you're going to multiply the first fraction by the reciprocal of the other. So that's just switching the numerator and the denominator. That will set you up to follow the same procedure for multiplication.
So thanks for watching this tutorial on multiplying and dividing rational expressions. Hope to see you next time.