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Adding and Subtracting Rational Expressions

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Today we're going to talk about adding and subtracting rational expressions. So we'll start by reviewing how to add and subtract numerical fractions, and then we'll do some examples adding and subtracting rational expressions.

So let's start by reviewing how to add and subtract numerical fractions. I've got 3/8 minus 1/8. When you're adding and subtracting fractions, you first need to make sure that the denominator is the same or that you have a common denominator. And we do. They both have a denominator of 8.

So once your denominators are the same, you can either add or subtract you numerators depending on what is in between the two fractions. So I can write this as 3 minus 1 over 8. 3 minus 1 is just 2, over 8. Notice that my denominator does not change. And 2/8 can be reduced to 1/4. So 3/8 minus 1/8 is 1/4.

For my second example, I've got 1/2 plus 2/3. So I notice here that I don't have a common denominator. So I need to start by multiplying each fraction by something to make a common denominator. So I'm going to multiply this first fraction by the denominator of my second fraction in both the numerator and in the denominator.

I'm going to then multiply my second fraction by the denominator of my first fraction, again in the numerator and in the denominator. So my first fraction, multiplying both the numerator and the denominator by 3 is going to become 3/6. And my second fraction, after multiplying by 2 in the numerator and the denominator will become 4/6. So I can see that I now have a common denominator of 6.

So 3/6 plus 4/6, I can combine my numerators and leave my denominator the same. So 3 plus 4 over 6. Adding my numerators, 3 plus 4 will give me 7. So my final answer is 7/6.

So let's do an example adding rational expressions. So I have 4x plus 1 over 3x plus 2x minus 5 over 3x. So similar with numerical fractions, I need to first make sure that my denominator is the same. And it is. My denominator in both fractions is 3x.

So now that my denominator is the same, I can go ahead and combine my numerators by adding. So this is going to become 4x plus 1 plus 2x minus 5 over 3x. My denominator still doesn't change.

I can combine any like terms in my numerator. And I see that I have 4x and a 2x. So added together, that's going to give me 6x. And then I have a positive 1 and a negative 5 added together is going to give me a negative 4. And my denominator will stay 3x.

Now there are no common terms or common factors between all three of my terms, so this is a simplified as my fraction can be.

So finally let's do an example subtracting rational expressions with an unlike denominator. So I have x minus 1 over 2x squared minus 3x plus 2 over 4x. So we first see that our denominators between the two fractions are not the same.

So before we can subtract them, we need to find the common denominator. And our common denominator can be represented by the product of all of the factors between our two denominators, or between all of the denominators. However we want the least common denominator so that our expressions do not get unnecessarily complicated.

So to find the least common denominator, I'm going to start by writing both of my denominators, listing them out by their factors. So 2x squared is the same as 2 times x times x. And 4x in factored form would be 2 times 2 times x. So I've written both of these denominators in their lowest factored form.

So to find my lowest common denominator, the LCD, I need to use any of the factors that appear in either of my denominators the most amount of times. So I see here for my 2x squared that I have a 2 and two x's. So I need to make sure that I have a 2 and two x's in my lowest common denominator, but I also need to incorporate the factors from my second denominator, 4x.

So for this denominator, I have two 2's multiplied together. Here I already have one. So I need to multiply by another 2. And then I have an x, but I already have an x as my lowest common denominator so I don't need to include an additional one.

So now simplifying this, 2 times 2 will give me 4, and x times x would give me x squared. So my lowest common denominator is going to be 4x squared.

Now that I know what I want my common denominator to be, I can multiply my first fraction by a factor so that this becomes 4x squared in the denominator and then do the same thing for my second fraction.

So to make 2x squared to be 4x squared, I'm going to need to multiply by 2. And since I do that in the denominator, I also need to multiply by 2 in the numerator, this entire numerator.

For my second fraction, to make my denominator of 4x to be 4x squared, I'm going to need to multiply that by x. Since I do that in the denominator, I'll need to multiply by x in the numerator. And again I'll need to multiply that by both terms in my numerator.

So now let's simplify this to see what we have. So I'll do 2 times x and 2 times negative 1, which is going to give me 2x minus 2 over my common denominator of 4x squared.

Then I will have minus. Again, I'll multiply x times 3x, which will give me 3x squared. And then x times 2, which would give me 2x, again over my common denominator of 4x squared.

So now that I've simplified, I can write as one fraction. So this will be 2x minus 2, subtracting this entire expression 3x squared plus 2x, over 4x squared.

And combining like terms, I have 2x and minus a positive 2x, which is just going to cancel. Then I have negative 2 and then I have my minus 3x squared. Those cannot be combined, so writing it in standard form, I'll have my negative 3x squared first and then my minus 2. My denominator is going to stay 4x squared.

And now I don't have any common factors between the terms in my numerator and denominator. So this expression is as simplified as it can get.

So let's go over our key points from today. A rational expression is a fraction in which the numerator and denominator are polynomials. To add or subtract rational expressions, write each fraction with a common denominator. Then add or subtract the numerators, keeping the denominator the same. This is the same process as adding or subtracting numerical fractions.

To find the common denominator, multiply both the numerator and the denominator of the first fraction by the denominator of the second fraction. Repeat this process with the second fraction.

So I hope that these key points and examples helped you understand a little bit more about adding and subtracting rational expressions. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.