Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Adding and subtracting rational expressions. First, we're going to factor the denominators. If necessary, we're going to find the least common denominator. Write an equivalent fraction with the LCD by multiplying the top and bottom by the same thing, hence multiplying by one. Add or subtract the numerators while leaving the denominator and then simplifying the results. So our first example 2Y and 2Y they are the same denominators. So we just combine the numerators and we get 8 / 2 Y and that we can see reduces to 4 / y because eight and two are each divisible by two. This next one, once again the denominators are the same, so X -, 3 Y plus X + 5 Y, all of that being kept over X + y. So we get 2X plus two y / X + y. We need to see if we can simplify that. So we're going to factor out A2 from the top and the bottom. The only thing they have in common is A1, but that makes it from a binomial to a monomial. So then the X + y S would cancel, leaving us 2 / 1 or just two. This next one, the denominators are not exactly alike, they're similar. So we're going to multiply the second one. We don't usually typically leave a negative in the denominator, so we're going to multiply by -1 / -1. When we do that, we get 2 / a + -5 / a. Now they have a common denominator, so 2 + -5 all over a -3 / a for our final answer. This next one, the denominators are almost the same, but they're not. So once again, we're going to multiply by a -1 / -1. So we're going to get three X + 1 / X -, 3 plus. When I multiply the -3, I get negative two X + 5 over, in this case X -, 3 when I distribute the -1. So once the denominators are the same, we literally just add numerators and combiner like terms. So we get an X + 6 in the numerator, X -, 3 in the denominator. That's as far as we can go because they're not monomials over monomials. If we wanted to look at it a little more in depth, we would factor out a one to make it a product or a product of two things, and we'd factor out a one in the denominator. And remember, once it is a monomial over monomial, it has to be exactly the same in the parentheses to cancel, and they're not exactly the same. So either of these last two are correct answers. This next one, we're going to do subtraction. I believe all the first ones were addition. Subtraction is going to be the same exact steps, except now we're going to subtract the second term. So we're going to start by getting common denominator. So we get 4X over X - 2 - -3 X plus 3 / X - 2. So when we combine these two numerators, we get four X - -3 X plus 3 / X - 2, SO4X plus three X - 3 / X - 2. A final answer of seven X - 3 / X - 2. Now sometimes we think about pulling this -1 and making it multiplied by the subtraction. So if we wanted to do that, the -1 times that subtraction a negative times a negative would make it a positive. So we would have had four X + 3 X -3 and we would have come up with the same answer. This next example, we need to get a common denominator. Well, they're currently monomials in the denominator. So that's a good first step. What's the common denominator between 3:00 and 12:00? And the answer is going to be 12 because three divides into 12 evenly and 12 divides into 12 evenly. Now between X ^2 and X here we're going to look for the highest exponent on the variable because X ^2 evenly divides into X ^2 and X evenly divides into X ^2. So we've got to get a common denominator. What did I do to three X squared to make it turn into 12 X squared? Well, we multiplied it the denominator by 4, but I can't just take four out of thin air. So if I'm going to multiply the denominator by 4:00, I've got to multiply the numerator by 4. So now we get 20 for 12X. What did I do to get 12 X squared? Well, I multiplied the denominator by X, so I've got to multiply the numerator by XI. Have to keep the equations balanced, so once I get a common denominator, now I just add and I get seven X + 20 / 12 X squared as my final answer. Can't Can't cancel XS, can't cancel the 20 and the 12 because it's not a monomial over monomial. If you wanted to look at it as a monomial over monomial, the only thing we could do to the numerator is factor out of one. Then once we get a parenthesis, it has to be identical in order to cancel. Example 7, we're going to get the highest exponents on all the variables, and then we're going to have to figure out what we need to multiply by to get that. So AB cubed. If I want a ^2 b to the 5th, I'm going to multiply by an A and AB squared. If I do the bottom, I have to do the top, so we're going to get 8 AB squared. The second one already has the common denominator, so we're going to leave it as a 5 S Our final answer is going to be 8 AB squared plus 5 / a ^2 b to the 5th. This next one, A - 6 is not a monomial, but there's also nothing A - 6 has in common. So we could think of it, if we wanted to, as 1 * a - 6 and as one times the quantity a + 4. Now to get the common denominator, we can see it's going to be one a - 6 and a + 4. So the first term has got to get multiplied by an A + 4 and the second term has to get multiplied by an A -, 6. Now one way to think about this when you're looking for that common denominator is to write down anything they have in common. There was a one in common, and then write down everything else that they don't have in common. So continuing on, we get 5A plus 20 plus seven A -, 42 all over A -, 6 * A + 4. Combining our like terms, we get twelve A -, 2212 A -22 A -6 A+ 4. And it looks like there's a two that can come out of the top, so six A -, 11 / A -, 6 A+ four and nothing else can cancel there. So that's our final looking at just a few more before we have you try some. We're going to have this one. The common denominator. I could factor a one out of the Y -, 2 to make it a monomial. So the common denominator is going to be 3 YY -2. So that fours got to get multiplied by y - 2 and the 5 has got to get multiplied by three Y. So when we distribute, we get four y -, 8 + 15 Y over three YY -2. So nineteen y - 8 / 3 YY -2. Can't do anything, can't go any further. The only thing I could do is factor out a one from the top. It wouldn't reduce this next one. This 4 is understood over a one, so the common denominator here is just going to be Z + 2. So I've got to multiply the four by Z + 2. And the other term is already over Z + 2, so we get four Z + 8 + Z -, 6 all over Z + 2, or five Z + 2 / Z + 2. Once again, that's as far as we can go. We can't cancel the Z's. We can't cancel the twos because currently it's not a monomial over monomial. If I wanted it to be a monomial, I have to factor out what they have in common, and both the numerator and the denominator, and then any parentheses would have to be exactly the same to cancel. So our last two examples are going to get a little longer and a little more complex. We're going to have to factor the denominators. So 2 numbers that multiply to give me 30 and add to give me 11 are going to be 5 and 6. And then two numbers that are going to multiply to give me 20 but add to give me 9 are going to be 4 and 5 S The common denominator here is going to be write down what they have in common 1st X + 5 and then write down anything that's not in common X + 6 X +4. Now this first terms numerators got to get multiplied by an X + 4 in order to get this common denominator. The second fractions numerator has got to get multiplied by X + 6 in order to get the common denominator. Now we're going to distribute X ^2 + 4 X -5 X -30 all over X + 5 X plus six X + 4, combining like terms X ^2 -, X -, 30 / X + 5, X plus six X + 4. Now are there 2 numbers that multiply to give us -30 and add to give us a -1? And the answer is yes, X + 5 and X - 6 multiply to give me -30 and add to give me that -1 X. So then we can see the X plus fives are going to cancel, leaving us a final solution of X -, 6 / X + 6 X +4. This next one's very similar. I think I'll go ahead and let you try it. Well, maybe I'll start this. Y ^2 -, 16 is y -, 4 Y +4. So this first term, the Y -, 3 is going to get multiplied by y + 4. The next one, the Y + 4 in the denominator, has got to get multiplied by a y -, 4, and the last term is already over the common denominator. So then we're going to have to foil things out and combine like terms in order to turn around and factor. So we get y ^2 + y -, 12 if I foil out the 1st 2 -, y ^2 -, 2 Y -8 if I foil out the second two plus the Y - 7 at the end. So continuing along the Y ^2 and negative y ^2 cancel y -. A negative 2 Y IS3Y plus AY is 4 Y -12 -. A negative 8 would give us -4 and -4 -, 7 is -11 So our final answer here is going to be four y - 11 / y - 4 Y +4. Thank you and have a wonderful day.