Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. When we add and subtract rational expressions, we need to make sure that our denominators are the same. So in this first example, 3 / 2 Y plus 5 / 2 Y, those denominators are the same. So the denominator stays the way it is, and we combine the numerators. So 3 + 5, which would give us 8, and then 8 / 2 Y is going to reduce. The top and the bottom are both monomials and they're both divisible by two. So we get 4 / y is the final answer. This next one, the denominators are the same, so we're just going to combine the numerators. And when we combine these numerators, we're going to realize we have two X + 2 Y over X + y. Now we don't know if we can reduce anything at this point because they're not monomials. To get it to be a monomial, we're going to factor anything they have in common. The top, they have a 2IN common, the bottom, they don't have anything in common other than A1. So we can always factor out A1. Now X + y / X + y are going to reduce anything over itself is always, always one. So we get 2 / 1, which is really just two. This one. The denominators are almost the same. We need to get this negative out of a denominator. If we want to get the negative out of the denominator, we're going to think about multiplying the second term by a -1 / -1. If we multiply by something over itself, it's really the same thing as multiplying by 1. So we're going to have the 2 / a + -5 / +a. Now that A is the same in the denominator. So we combine our numerators. 2 + -5 is going to give us -3 / A this next one. The denominators once again are almost the same, but they're not the same. They're exact opposites as far as the signs go. So I'm going to choose to multiply by -1 / -1 in one or the other of the two fractions. I'm going to choose the second one because I personally like to have my variable positive. If I do this, I haven't changed that first fraction, so it's going to be three X + 1 / X - 3. But the second fraction, all my signs have changed negative two X + 5 / X - 3. The denominators are now the same, so we just combine the numerators. Once we've done this, we're going to have X + 6 / X -, 3. Those are not monomials. So we could factor out a one on the top, we can factor out a one on the bottom. We'd see that those two parentheses are not exactly the same. So we can't reduce anything in this problem. So our final answer is going to be X + 6 / X -, 3. Now you don't need the one in the parentheses, but you need to be very, very careful not to think that you can cancel those XS. Can't cancel the XS because they weren't a monomial back here in the pre factored form. Once it's factored, the parentheses are not exactly the same. So that's our final answer. This next one, the denominators are currently exactly the same, but it is a subtraction problem this time. So it's going to be that first numerator minus every term in the second fraction or the 2nd numerator. So you could think about going ahead and distributing out that negative first. When we subtract, that subtraction really changes all of the signs in the parenthesis. Then if we combine our like terms, we're going to get two X ^2 + 8 X -11 / 2 X -3. Are there any 2 numbers that would multiply to give us -22 and add to give us 8? And the answer is no. So the top can't be factored any more than pulling out A1, the bottom can't be factored, and once we just pull out A1, we can see that the parentheses are not identical. Hence, that's our final answer. Be very, very careful. You can't cancel the twos. You can't cancel the X's. The parentheses have to be exactly the same, and it has to be a monomial over a monomial. This next one, the denominators are not exactly the same, but they're close. So we're going to take one of these two terms and multiply by a -1 / -1. When we do that, what it really does is it changes all the signs in the fraction. So negative three X + 3 / X - 2. Now the denominators are the same and we're going to subtract. When we subtract, we can think of that really as changing all the signs. So we're going to get a positive 3X and a -, 3. So we're going to get seven X -, 3 / X -, 2 addition and subtraction of rational expressions when we have unlike denominators. So our first thing that we have to do is we need to look for a common denominator. And common denominator is going to be a number that both those denominators can evenly divide into. There's lots and lots and lots of ways to do this. One way is to think about what do these two denominators have in common? They each have an 8IN common. So if we wrote down what they have in common, then we also have to write down everything that they don't have in common. So the common denominator for this problem is going to be 48. And if we think about 48 / 16 is an even number, it's 348 / 24 is a whole number and it's two. So when we look here, we had a 2 * 8. We don't have a three to get that 48. Well, I can't just pull a three out of thin air. So if I'm going to have to multiply the bottom by three in order to keep the equation balanced, we're going to multiply the numerator by three because 3 / 3 is really the same thing as one. So this new denominator or new numerator turns into three times 3X or 9X. In this other fraction we have three and eight, but the common denominator was 2 * 3 * 8. So I need a two in the bottom. But if I'm going to do a two in the bottom, I have to also do A2 in the top. So now we get 2 * 7 X squared or 14 X squared. Once the denominators are the same, then we just combine our numerators. If they're not like terms, we can't do much. We could factor an X out and get fourteen X + 9 / 48. Correct answer either form fourteen X ^2 + 9 X or X * 14 X plus 9 / 48. Both of those are over 48. We usually do our terms in descending order, so the 14 X squared should come before the 9X. This next example, we need to get a common denominator. So we could think of as three and 12 is really three times four X ^2 X * X and the other had just a single X. So they have a three in common. They also have an X in common. Once I write down what they have in common, the three and the X were in common, I have to also write down everything else that was there. So we need to write down the four and the X. Also, if I have 3X times 4X, that's really going to give us a common denominator of 12 X squared. And if we think about that, twelve X ^2 / 3 X squared would be 4. Twelve X ^2 / 12 X would be X. So they both divide evenly in. If we look at this three X squared, we need it to multiply the bottom by a four to make it turn into a 12 X squared. But if we do the bottom by four, we have to do the top by 4. So we're going to get 4 * 5 in that first numerator, or 20. If we look at the second term, we have a 12X and we want it to turn into a 12 X squared. So we're going to multiply the top and the bottom each by X. So we're going to get 7X. Once we get the denominators the same, we just add the numerators. I'm going to put that 7X in front because we go in descending order, so seven X + 20 / 12 X squared. This next example, one of the easier ways to do it if we just have variables is we need the variable with the highest exponent. So the common denominator here is going to be a ^2 b to the 5th. If we have a ^2 b to the 5th, what did we have to multiply this first one by? To get a ^2 b to the 5th? We needed 1A and AB squared. And if I do the bottom by AB squared, I've got to do the top by AB squared. So we're going to get 8 AB squared. The second term already had the common denominator, so when we combine them, we get 8 AB squared plus 5 / a ^2 b to the 5th. And that is our final answer for that one. Let's look at a few more examples. Here. A -, 6 and a + 4. The denominators didn't start out as monomials. I could pack factor just a one out of a -, 6 and just a one out of a + 4. Now those denominators are monomials. What do the two denominators have in common? They only have a one in common. So we're going to have 1 * a - 6 * a + 4 as our common denominator, and that's going to be the denominator for both the fractions. If we look at that first fraction, it only had an A - 6 originally. So we're going to need to multiply the top and the bottom each by the a + 4 because a + 4 / a + 4 is really one. So that would have given us what we started with the 5 / a -, 6. This next one didn't have an A -, 6 originally. So we're going to multiply the top and the bottom each by a -, 6 to get the common denominator. Now, once the denominators are the same, we're going to combine our numerators by distributing out the numerator portion and then combining like terms. So we're going to get 5A plus 20 plus seven a -, 42 or 12 A -22 / a -, 6 A plus four. We could take out a 2 on the top, which would give us six a -, 11. Now that there's monomials on both the numerator and the denominator, we can see that nothing will reduce. And that's our final answer. You could leave it as twelve a -, 22 if you wanted, but usually we leave it as a monomial so that we can visibly see that it can't reduce any further. The next example, the 4 / 3 Y is currently a monomial, but the Y - 2 is not a monomial. So we're going to factor out A1 and then we're going to realize that they don't have anything in common. So the common denominator is going to be three y * y -, 2 and three y * y -, 2. This first term has to have AY -2 multiplied in the top and in the bottom. The second term we have to have a three Y multiplied in the top and the bottom. Once those denominators are the same, we're just going to add the numerators. So we're going to get four y -, 8 + 15 Y. That's going to give us nineteen y -, 8 three y * y -, 2. The top doesn't have anything other than a one that can come out. So when we look at this, we realize in monomial over monomial that nothing can cancel. The next example, we have X - 5 * X + 5 as the common denominator because the second term, if we thought about pulling out A1, they'd have an X - 5 in common and we'd have to put an X + 5 in the top of the second term to get the common denominator. Once we get our common denominator, we're going to combine the numerators. So we're going to get 3X minus four X ^2 - 20 X. Be careful and remember that there's that negative or that subtraction way out here in front. That negative goes with the 4X when I distribute it. So -4 X times X and negative four X * 5. We're going to combine our like terms so we get negative four X ^2 -, 17 X over X - 5 X +5. These each now have a negative X in common, so we'd get four X -, 17 if we factored it over X -, 5 * X + 5. This next 14X and 6X the common denominator is going to be 12X, so we need to multiply the first numerator by three, and we need to multiply the 2nd numerator by A2 in order to get the common denominator of 12X. So nine X -, 6 -, 6 X -2 all over 12X. Combining our like terms, we get three X -, 8 / 12 X. We could factor out a one because that's all that that numerator has in common. And then we can see that there's nothing exactly the same that will cancel because remember, if it's got a parenthesis, it's got to be exactly the same in the parenthesis to cancel. So that's our final answer. The next example, we need to factor the denominators because they're not monomials. So we're going to get X + 5 * X + 6, and we're going to get X + 4 * X + 5. To get the common denominator, we list what they have in common, which is the X plus fives. Then we list everything that's not in common. Once we get those common denominators, we have to figure out what's not already in the numerator or in the denominator that has to get multiplied by the numerator. So this first one we need an X + 4 in its numerator, and the second one we need an X + 6. So we're going to distribute the numerators and combine them X ^2 + 4 X -5 X -30 over X + 5 * X + 6 * X + 4. We need to figure out if we can factor X ^2 -, X - 30, and if we can, then we're going to see if we can reduce it with the denominator at all. Well, that's going to be X + 5 * X - 6 in the numerator, which means that X + 5 will cancel. So our final answer for this one is going to be X - 6 / X + 6 X -4. Let's look at just a couple more examples. This next one, we're going to do 3 terms instead of two. The Y ^2 - 16 is y + 4 Y -4 because it's the difference of squares y - 4. We could factor out A1 to show that we understand it's really a monomial y + 4. We do the same thing. So this first fraction that y - 3 in the numerator needs to get multiplied by y + 4 in order to be over the common denominator. The second term has got to have that y + 2 being multiplied by y -, 4 for it to be over the common denominator. And the last term was already the last fraction was already over the common denominator. So we're going to have to do lots of foiling. We're going to get y ^2 + 4 Y -3 Y -12 minus. I'm going to put this in parentheses so that I can do it a little bit step by step and maybe not have so many sign issues at the end. Y ^2 - 4 Y plus two y - 8 + y - 7. So now that subtraction really says to. Think about multiplying it through by a -1. So y ^2 + 4 Y -3 Y -12 -, y ^2 + 4 Y minus two y + 8 + y - 7. So now we're going to combine our like terms. Our y ^2 and our negative y ^2 are going to cancel four y - 3 Y is y + 4 Y Excuse me, +4 Y is five y -. 2 Y IS3Y plus another Y is 4 Y. And then when we do our constants -12 + 8 is -4, negative 4 - 7 is -11. So four y - 11 / y + 4 Y -4. Be careful, the fours won't cancel, the YS won't cancel. If you don't see that, make that numerator into a monomial by factoring out of one and we get y + 4 Y -4. OK, the last example t ^2 - 1 is t - 1 * t + 1. So our common denominator is going to be T times t - 1 * t + 1 for these three fractions, and we have to have that common denominator for all three of them. So the first term we're going to have to multiply the numerator by AT so we have 4T times T The next fraction we're going to have to multiply by t - 1 and t + 1. And the last fraction we're going to need to multiply by T and t -, 1. When we multiply this out, combine our like terms, we're going to get four t ^2 -, 2 T squared. If we put in our parentheses, that would be two t ^2 -, 2 plus two t ^2 - 2 T all over that common denominator. If we distribute the negative, we'd get 4T squared minus two t ^2 + 2 plus two t ^2 -, 2 T all over the common denominator. Now if we just combine our like terms, 4T squared minus two t ^2 + 2 T squared is four t ^2 -2 T +2, and that numerator could have a 2 come out of it. So we'd get t ^2 2 T squared minus t + 1, and that will actually factor into two T Oh, Nope, that doesn't factor. So that's actually our final answer right there. Two times the two t ^2 - t + 1 / t * t - 1 T plus one. OK, on this one, we need to factor the denominators. So we're going to get three y * y and plus four and -1 over here we're going to get three y + 4 and y -, 2. So the common denominator, what they have in common, three y + 4. Then we also have to write everything they don't have in common. That 7's got to get multiplied by y -, 2 and the nine y + 2 is going to have to get multiplied by the Y -, 1. Now when we do this, we want to make sure to foil things out or distribute. So seven y -, 14 plus nine y ^2 -, 9 Y +2 Y -2 all over that common denominator of three y + 4 Y -1 Y -2. If we combine our like terms, we're going to get nine y ^2 +7 Y -9 Y and +2 Y. The YS will cancel -14 - 2 is -16 / 3 Y +4 y - 1 Y -2. Now that numerator is not a monomial and it is the difference of two squares. So three y + 4 * 3 Y -4 and three y + 4 Y -1 Y -2 still in the denominator. So three y + 4 is in top and bottom and it will cancel. So our final answer here is going to be three y - 4 / y - 1 * y - 2. On this next one, the common denominator. If we factored out of one, we could see that t - 1 and t + 1 really don't have anything in common. So the common denominator is t - 1 * t + 1. So that 3T is going to get multiplied by t + 1 and that AT is going to get multiplied by t -, 1. So when we do this, we need to make sure to distribute and we're going to get three t ^2 + 3 T minus eight t ^2 + 8 T Remember to distribute the -3 when we distribute, if we combine our like terms here, we're going to get negative five t ^2 + 11 T over t - 1 T plus one. So on the top, we could take out a negative T, which would give us five t - 11 / t - 1 T plus one. Thank you and have a wonderful day.