Hi, and welcome. My name is Anthony Varela. And today we're going to add and subtract rational expressions.
Well, we're going to start with a review of adding and subtracting numeric fractions because rational expressions are just algebraic fractions. So if you can understand how to add and subtract using numbers, we can apply that to when there are variables.
Now finding a common denominator is incredibly important for adding and subtracting numeric fractions. And that holds true for rational expressions as well. So we'll practice finding the common denominators of algebraic fractions.
So when we're adding two fractions, and if they have common denominators, that makes it pretty easy to add. Like in this example here, we have 3/8 plus 2/8. I can just keep that common denominator. That doesn't change at all. That's still 8. And we just add our numerators. 3 plus 2 is 5.
So if we have a common denominator, it makes it very easy. We just keep that denominator throughout the whole equation and just add the numerators.
So if we have two fractions that don't have a common denominator, we'd like to rewrite these fractions so that they do. Now one thing that I can do is just multiply the denominators to make common denominators. So I can multiply 8 by 3 and multiply 3 by 8. So we can see that their denominators are both going to be 24, or 3 times 8.
But I have to adjust my numerator. So in 3/8, because I multiplied the denominator by 3, I need to multiply the numerator by 3. And in 5/3, because I multiplied the denominator by 8, I have to multiply the numerator by 8.
So we can see how we're not actually changing the value. We're just creating a common denominator. So I know that my denominator is going to be 24. And here I have 9/24, plus 40/24.
I can keep that common denominator and add 9 plus 40 to get 49 over 24. So now we're going to get into the hard part, when there are variables. So how can we add and subtract algebraic fractions?
Well, we'll start with a simpler case where we have a common denominator. So I have 2x plus 1 over 3x. And I'm subtracting 5 minus x over 3x.
So I'm going to keep my common denominator of 3x. That's not going to change. Now I'm just going to subtract my numerators. So I have 2x plus 1. And I'm going to subtract 5 minus x.
Now, subtraction is a little bit more difficult than addition because we have to account for everything in here being subtracted. So I'm actually going to distribute the negative 1 that's implied with the subtraction.
So rewriting this then, I have 2x plus x. So that gives me 3x. Positive 1 minus 5, so I get a negative 4. And my common denominator is 3x.
And this is as simple as it gets because there are no common factors between the numerator and denominator that can be canceled. Remember, we can't just cancel out this 3x because the 3x on top has a minus 4 attached to it.
So now we're going to get to the truly difficult case when we have two algebraic fractions that we'd like to combine and they don't have common denominators. Well, we need to create our own common denominators.
And it can get a bit tricky. It's not just as straightforward as multiplying our two denominators because you might end up with some common factors on the top and the bottom, which can be canceled. And that can get a bit difficult or messy with the variables.
So what we're going to do is find the least common denominator. So that already cancels out any common terms from the top and the bottom of our fraction. So how can we find the least common denominator?
Well, in order to find the least common denominator, the first thing we're going to do is factor the denominators as much as we can. So looking at our first denominator, 5x, that factors into 5 times x. 5 is a prime number. We don't have to write the 1. So we have 5 times x.
Now taking a look at 3x squared, breaking these down into its factors, we have 3 times x times x. So one factor of 3, two factors of x.
Now that we've factored this out as much as we can, we are going to take a look at the greatest number of times each factor is used in these factorizations here. And then we're going to multiply those together to make our lowest common denominator.
So this is what I mean. So taking a look at 5, 5 is used at most one time, because we see it here once. We don't see it here at all. So we're going to include one factor of 5 in our lowest common denominator.
All right, well, now let's take a look at 3, our other number here. I see it used once in this factorization, zero times here. So at most it's used once. So we're going to include it one time in our lowest common denominator. So so far, our lowest common denominator has one factor of 5 and one factor of 3.
Now what about x? Well, x is used once here, but it's used twice here. So two would be the greatest number of times the factor of x is used. So we're going to include two factors of x in our lowest common denominator.
So our LCD is 5 times 3 times x times x. So I'm going to rewrite this fraction to have a common denominator of 15x squared. That's just 5 times 3 times x times x.
So we'll bring down what we have in our numerator so far. But taking a look at what we had to multiply to 5x to get 15x squared, well, we had to multiply a 3 and an x. We already had 5x, so we need to multiply 3x. So now these two fractions are equivalent.
Now we're going to rewrite our second fraction to have a denominator of 15x squared. So I'll bring down what we have so far, 2x plus 3. But what do we have to multiply it by 3x squared to get 15x squared?
Well, 3x squared already has factors 3x and x. So we just need to multiply this then by 5. So here are two fractions written with a common denominator.
So now before I add these, what I'm going to do is just simplify the numerator. So I'm going to distribute 3x into x plus 1. So that would be 3x squared plus 3x.
And I'm going to do the same thing here. I'm going to distribute 5 into 2x and 3. So that would give me 10x plus 15.
Now because these have common denominators, I can write this as one fraction with my denominator of 15x squared. And now I just need to add all of these parts.
So 3x squared plus 3x plus 10x is 13x plus 15. So that is our sum of our two fractions that we started with that didn't have common denominators.
So let's review adding and subtracting rational expressions. Well, our common denominator is key to adding or subtracting our rational expressions. We can just keep that common denominator and add or subtract across the numerators.
If we don't already have a common denominator, we can make one by finding the lowest common denominator. Factor all denominators completely, find the greatest number of times each factor is used when you were writing down all the factors, multiply them together, and that's going to give you your common denominator. So you can always follow these steps.
So thanks for watching this tutorial on adding and subtracting rational expressions. Hope to see you next time.