For this lesson, we're going to try our hand at adding rational expressions. We're going to draw upon our knowledge of adding fractions to figure that out. For example, if we wanted to add 3 11ths to 2 11ths, we would just add the numerators and keep the denominators the same. We would get a total of 5 11ths. Keep in mind we can only add or subtract two fractions whenever the denominators are like. Keeping this in mind, I want you to try and add these rational expressions together. This last one isn't addition though, it's subtraction. So you want to make sure you subtract the numerator. You'll also need to use your knowledge of simplifying rational expressions for these two problems. Try your best, and don't worry if they're not right on the first try.
Here are the correct answers. Fantastic work if you got at least one of those correct. If either of these problems are giving you trouble, then watch the solution and we'll figure it out together. To add these two fractions together, we simply add the numerators. 3 n and 2 n equal 5 n. We leave the denominator alone, so we have 5 n divided by 11. This problem is exactly like the one we saw before. It's just we needed to combine the like terms and our numerator first. For the second problem, we want to add the numerators a and b together, and keep the denominator the same. Since we have a plus b divided by a plus b, we know those reduce to 1. And finally, for this third one, we want to subtract a y from this numerator. So we'll have 2 y minus 1 minus 1 y, we subtract the like terms to get y minus 1 for our numerator. Now we want to factor our denominator to see if it can reduce with our numerator since our numerator has changed and it turns out that's a great move because we can factor and reduce and equivalence of 1. Y minus 1 divided by y minus 1 equals 1. This leaves us with the answer of 1 divided by y plus 2. If you got the first two, excellent work, and I know this one might of given you some trouble.
But what if we wanted to add these 2 fractions together? What do you think we could do? Well, adding these 2 rational expressions together is very similar to adding 2 fractions together. If the denominators are different, we need to change each fraction. By finding the lowest common denominator first. We know the lowest common denominator would be 90, the product of these factors. We multiply 1 10th by nine over nine, which is the missing factor from ten. 1 10th by 9 9ths is 9 90ths. For 1 18th we multiply by five over five. This is the missing factor for 18. So 1 18th time 5 5ths is 5 90ths. Now we add these 2 fractions just as we learned before. 9 plus 5 equals 14 and we have 14 90ths. These numbers are both even and so we can reduce them by 2. So these are a couple of fractions.
Now, that we've remembered how to add two fractions together, I want you to apply this method to figure out the LCD for these two fractions. What's the lowest common denominator?
Two K S squared is the lowest common denominator, great work if you found it. If you're wondering why that's true, it's because these two numbers share a common factor of two. Remember these variables are the placeholders of numbers so two times K would be a number and two time S squared would really be a number as well. We don't know what K is, but it's the other factor for this denominator. And the same is true for s squared. It's the other factor for 2 s squared. We multiply all these together to get 2 k s squared. Now I know I did a little bit of reordering in this situation, but we usually list the coefficient or number first. Then we list the variables in alphabetical order.
Now that we know that the LCD is 2ks squared. What should we multiply each of these fractions by in order to get new fractions with denominators of the LCD? Write your answers in these boxes.
Well, we should multiply the first fraction by s squared divided by s squared.This would give us s squared over 2ks squared. I know I multiply this first fraction by s squared divided by s squared. S squared is missing from the new denominator or our lowest common denominator. For the second fraction, we only need to multiply by k over k. 2s squared times k would give us our lowest common denominator. 2k s squared. Keep in mind that we're really just multiplying by a form of 1 in both of these cases. We're not changing the value of the fractions, we're just changing what they look like.
So, just like when adding fractions, we start out by finding the LCD or the lowest common denominator. Next, we multiplied each of these fractions by the missing factors of the lowest common denominator. That was the same for our actual numbers. Now, we're ready for the last step. Let's find the actual fractions that are equivalent and then find the sum. So, what are these two equivalent fractions? And then, what I want you to do is to find the sum of the two fractions and put that here. Good luck.
For the first fraction we'll have s squared for a numerator and 2 k s squared for the denominator. For the second fraction we'll have k for a numerator and 2 k s squared for the denominator. And now we're ready to add these two fractions together. We add the numerators together, just like from before and then keep the denominator the same. Be careful that you don't multiply these two numbers together. We really just have s squared plus k. We don't multiply them, since we're adding.
So whenever we want to add rational expressions together we should find the lowest common denominator of our fractions. The lowest common denominator is the product of all of the different factors that appear in each denominator. Keep in mind that we want to take the highest power of any factor that appears in a denominator. We're not going to jump into the entire process yet. Let's just get comfortable with this first step of finding the lowest common denominator.
What LCD or lowest common denominator would we need in order to add these two fractions together? Now, we aren't going to actually add the two fractions together since we're just getting comfortable with the first step. What do you think the LCD would be? Write it here.
If we had a binomial or trinomial in these denominators, we would want to factor them first. But since these are monomials, we can use our factor tree method to figure out what they share. I know 9 and 15 share a common factor of 3, and the a's share a common factor of 1a. The highest power of b that's in common is b squared. So 3ab squared is my greatest common factor. The other factor for this denominator is 3a squared. If we multiply these two together, we'll get our original denominator. Whereas for this fraction, the missing factor is 5b squared, if I multiply these two together we'll get 15, 3 times 5, a, and then b squared times b squared or b to the fourth. We multiply all these factors together to get our LCD. So our lowest common denominator is 3 times 3 times 5, which is 45 and then a squared time a which is a cubed, and b squared times b squared which b to the fourth. I said earlier that you want to use the highest power of any factor for the lowest common denominator. Notice that we have a cubed in a. A cubed is the highest power in either denominator. So we want to make sure that appears in our LCD. Likewise between b squared and b to the fourth, this fourth power is higher, so we know b to the fourth needs to appear in our LCD. This is a quick way you can check to make sure your variables or your factors are correct.
Let's try another problem where we find the lowest common denominator. You'll need to factor these denominators first, since they're not just monomials. As a hint, your LCD will contain a constant and two binomials. And you'll want to make sure you leave it in factored form. I know you might not have seen this before, but I want you to give it a try. If you get stuck, there's always this solution.
The LCD is 2 times u minus 3 times u plus 3. Fantastic work if you figured that one out. Again, I don't expect you to get that right on your first try, or even maybe at all. Let's see how we do it. We know u squared minus 9 is a difference of two perfect squares. So we can factor this as u plus 3 times u minus 3. So we can rewrite this first fraction like this. For the second denominator we have a common factor of 2. So we can factor that out. Now that we have factored denominators, we can use our factor tree method. These two denominators share a common factor of u minus 3, and the other factor for this one is u plus 3, while the other factor for this one is positive 2. And there it is. We multiply all these together to get 2 times u plus 3 times u minus 3, our LCD.
Alright, here's our third and last example of trying to find the LCD. Remember we're not quite adding these two rational expressions together yet. We just want to get a common denominator since these two denominators are different.
The lowest common denominator for these three fractions would be p minus 1 times p minus 2 times p plus 3, fantastic work for finding that one. First we factor this denominator to p minus 2 times p minus 1, we find factors of 2 that sum to negative 3. For our second denominator, we find factors of negative 6 that sum to positive 2, which are positive 3 and negative 2. Once our denominators have been factored, we're ready to use our factor tree. So we have p minus 1 times p minus 2 times p plus 3. This is our lowest common denominator.
So why do we need to find the lowest common denominator? Well remember that we have to find a common denominator in order to add or subtract fractions. In our very first example, we needed to have 90th in order to add these two fractions together. In the second example, we needed to have a common denominator of 2ks squared in order to add these two fractions together. Once we have a lowest common denominator, we multiply the numerator and the denominator in each fraction by the factors that are missing to change our fraction into an equivalent form. So for 18, it was missing a factor of five. So we multiplied by fraction by k over k. Since k is the missing factor. From 2s squared. When we multiply by k divided by k, we're really multiplying by 1, and we're changing our first fraction into its equivalent form. So let's go back to our third example of finding the LCD. We found the lowest common denominator for these two fractions, and it was p minus 1, times p minus 2, times p plus 3. Remember we want to change these two fractions to an equivalent form. So here's your question. What should we multiply each of these fractions by to get an equivalent fraction with a lowest common denominator of this?
We multiply this first fraction by p plus 3 divided by p plus 3. The denominator here is missing a factor of p plus 3. This is the one factor we need in order to change our denominator into the LCD. For our second fraction we multiply by p minus 1 divided by p minus 1. And this denominator, p minus 1 is the missing factor we need in order to get the LCD. Now we multiply these two numerators together and these two denominators together this gives us our new first equivalent fraction. We do the same process for the second fraction to get this, these two denominators may look different but they really the same. We know the order of the factors don't matter because its multiplication. Now that the two fractions have like denominators, we can simply add the numerators together. We add the like terms. The p's and the constants. 1 p and 1 p would equal 2 p and positive 3 and negative 1 would equal positive 2. Now that we're here, we should ask ourselves, are we done? Well, we added two things in a numerator together, so we have a new numerator entirely. We want to see if we can factor it if possible, and it turns out we can. We can take out a 2, to have 2 times p plus another factor of p plus 1 does not appear in the denominator. So the answer is yes, we're done, and here's the addition of these two rational expressions. Now this is some algebra.
So here's our guide, our steps that we could follow to add rational expressions together. First, we want to factor the denominators and find the LCD. Next, we multiply each fraction by the missing factors of the LCD to get equivalent fractions. Finally, we add the numerators together and then we check if we can simplify this by cancelling any factors. And again, don't make the mistake of canceling terms. We can't cancel these since this is addition in the numerator. We can only cancel factors.
Let's try adding these two rational expressions together an see if we can get the correct answer. We're not going to do all of the steps at once instead we're going to break it down into three pieces let's start with the first one. What do you think would be the lowest common denominator for these two fractions? Write your answer here.
We find the lowest common denominator by factoring each of these denominators first. The first denominator will have factors of x plus 2 times x minus 6, since factors of 12 that sum to negative 4 are positive 2 and negative 6. The factors for x squared minus 36 would be x minus 6 and x plus 6. This is just the difference of two perfect squares. We can see that both of the denominators share a common factor of x minus 6. So this is the factor they share, and we list it in the middle. The other factor of this denominator is x plus 2, and the other factor of this denominator is x plus 6. Finally, we just multiply all these factors together to get our lowest common denominator, right here.
We've completed step 1 and we found the lowest common denominator for these two fractions. Now we're ready to make equivalent fractions for each of these. What do we need to multiply this fraction by and this fraction by to get our new equivalent fractions? You can write the factors in these boxes. Good luck..
We need multiply the first fraction by x plus 6 divided by x plus 6. We know that because this denominator is missing a factor of x plus 6. If we multiply the numerator and the denominator by x plus 6, we're multiplying by a form of 1, and we changed this denominator into our lowest common denominator. For the second fraction we need to multiply by x plus 2 divided by x plus 2. This denominator is missing our factor of x plus 2 to form the LCD. If you got these correct, excellent thinking.
Now that we're changing each of these 2 fractions, I want you to multiply these numerators together and write your answer here. For the denominators, we don't want to multiply them out. We just want to list the factors. This is because we're going to add the numerators in the end. We'll refactor that if possible and see if we can cancel any factors in the numerator and denominator of our final answer.
Six x plus 36, and five x plus ten, are our new numerators. Great work if you found those two. We know we're really just distributing the six to the x and the six. So six times x is six-x. And six times six is 36. For this new numerator, we're distributing the five times x, which is five-x. And 5 times 2 which is 10. So, 5x plus 10 is our new numerator.
First we found the LCD and then we changed each fraction by multiplying by the missing factors of the LCD. Now that we've converted this fraction and this fraction into new fractions that have common denominators. What would the sum be? Write the numerator here and the denominator here.
we would get 11X plus 46 divided by the factors x plus 2 times x minus 6 times X plus 6. Fantastic work if you just added these numerators together. When we add the numerators we just want to add the like terms, like 6X and 5X Which would equal 11x. Likewise we would add the constant terms. Positive 36 and positive 10 equal positive 46. So here's our numerator and here's our denominator.
Well, the answer would be no. When we look at 11x and positive 46, we can think about the greatest common factor between them. That greatest common factor would just be 1. Because there's no other factor of 11x plus 46 in our denominator, we can't cancel this factor with anything down here. This is another reason why we want to leave the denominator in factored form. At the end, we should check to see if this is factorable and cancel any factors that appear in the numerator and denominator.
But what if our sum, instead, was 11 x plus 22? What do you think our final answer would be this time? Now, I know this wasn't the sum for our last problem, but I want you to see if there's an extra step you need to do at the end. Sometimes you need to simplify further. Can you in this case? Write your answer here.
It turns out we can. The numerator would be eleven and the denominator would be X minus six times X plus six. Great job, if you've figured that one out. When I look at this numerator and these two terms, I can see they share a common factor of eleven. If we factor eleven out of our numerator, we'll get eleven times X plus two. Now we can see this X plus two is a factor and the numerator. Add in the denominator, which means we can simplify. These factors cancel to 1, and we're left with 11 divided by x minus 6 times x plus 6.
Alright, now you try one on your own. I want you to add these two rational expressions together and remember to check to see if you can simplify your numerator and your denominator in the end. Make sure your numerator is multiplied out in the end, and make sure the denominator is in factored form. Have fun and good luck.
This was our final answer. Excellent work if you found it. We start by finding the lowest common denominator, which means that we have to factor this denominator and this one first. Factors of 24 that sum to positive 10 are positive 6 and positive 4, while the factors of negative 12 that sum to positive would be y plus 6 times y plus 4 times y minus 3. Next, we multiply each fraction by the missing factors of our lowest common denominator. For the first denominator, it's missing the factor y minus 3. So we multiply the numerator and denominator by y minus 3, since we're really just multiplying by 1. For the second fraction, we need to multiply by y plus 6 divided by y plus 6. When we multiply each of these fractions by a form of 1, we get two new equivalent fractions. We multiply these two numerators together, or these two binomials together, to get y squared minus 6y plus 9. And then here again, we multiply these two binomials together to get y squared plus 12y plus 36. We multiply these out, since we're adding our numerators in the end. The denominators, however, we can just leave in factored form since these won't change. When we add the like terms of our numerators together, we get 2y squared from 1y squared and 1y squared. We get positive 6y, from negative 6y and positive 12y. And we get 45 from positive 9 and positive 36. And finally, we want to check to see if this is factorable, to see if we can cancel a factor in the numerator and denominator. We find factors of 90 that sum to positive 6. When I look at the positive factors of 90, these are the factor pairs. None of these add to positive 6. So I know this numerator is not factorable. Since we cant factor this numerator, we actually have our final answer. None of these add to positive
For our third practice problem, I want to you to try to add these two rational expressions together. Write your answer here.
This was the correct answer. Fantastic job if you found it. Now if you got stuck along the way, it might of been at this denominator that gave you trouble. Let's see how we can find our lowest common denominator first. This denominator factors to x plus 2 times x minus 1. And then this denominator is actually the difference of two perfect squares. So we can factor it as X plus 1 times X minus that in the middle for our factor tree method. The other factor's on the outside and then we multiply all of them together to get our lowest common denominator. X plus 2 times X minus 1. Times x plus 1. Next we want to turn each of these fractions into an equivalent fraction with the lowest common denominator. This first denominator is missing the factor x plus 1. So we multiply the numerator and denominator by it. The second fraction is missing the factor x plus 2. For this numerator we're multiplying 2 binomials together and we get x squared minus x minus 2. For the second fraction we're[INAUDIBLE] this 3, the quantity x plus added our numerators together we have a new numerator. And we should always ask ourselves can we factor this? If we can factor this then we'll be able to simplify another factor in the numerator and denominator. But in this case, this numerator can't be factored. There are no factor pairs of 4 that sum to positive
To subtract this fraction, we're going to distribute this negative sign to every term in the numerator. We know subtraction is the same as adding the opposite. And in this case, we only had to change the sign of one term, the positive 3. If there were more terms in the numerator, we would want to change the sign or distribute the negative 1 to each of those terms. Now, we're in familiar territory. We know how to add two rational expressions with different denominators. First, we'll find the LCD, create equivalent fractions, and finally, add the numerators together and check to see if we can simplify. So, you give this one a shot. What would be the final answer?
This would be our solution, great work if you found it. We want to start by finding the lowest common denominator, so we should factor each of these denominators. We factored this denominator to be m plus 1 times m minus 5. This denominator, we'll need to use factoring by grouping. You need to find factors of 30 that sum to negative 13. Those factors are negative 3 and negative 10. They multiply to give us positive 30, and they add or sum to give us negative multiply our first fraction by the missing factor of two m minus three, whereas we multiply our second fraction by the missing factor of m plus one. And remember, we're really just multiplying by a one here and a one here. We're changing these fractions into equivalent forms. We distribute m to 2m minus 3 to get the first numerator, 2m squared minus 3m. Next, we distribute the negative 3 to the m plus 1. We'll get negative 3m and negative 3. Now, we want to add the two numerators together and keep our denominator the same. The like terms in these numerators are negative 3m and negative 3m. So we have 2 m squared, minus want to find factors of -6, that's sum to -6. Well there are no factor pairs of -6 that sum to you -6. So we're finished. So really subtracting rational expressions is not that different from adding them. We just want to make sure we remember to distribute the negative sign.
Okay, try this one on your own. Subtract these two rational expressions and write your numerator here and your denominator here. Good luck.
To subtract the second rational expression we really want to add the opposite. We want to distribute our subtraction sign to each term in the numerator which gives us a negative 1. Now we factor each denominator and try to find the lowest common denominator between the two fractions. The lowest common denominator is x minus 1 times x minus 1 times x plus 1. We know this since these two denominators share a common factor of x minus 1. The other factor for this denominator is x minus 1. And the other factor for this denominator is x plus 1. We multiply each fraction by the missing factors of the lowest common denominator. For the first fraction, we distribute 6x to the quantity x plus 1. We'll get 6x squared plus 6x. For this second fraction, we'll have negative 1 times the quantity x minus 1. We really just changed these two signs. So we'll have negative x plus 1. Now that we have two fractions with common denominators We can add the numerators together. We just add the like terms. When we add the like terms of 6x and negative x together we'll get 6x squared plus 5x plus 1. Now remember, we have a new numerator here. And let's see if we can factor it. We want factors of positive 6 that's sum to 5. Well yes that would be 3 and 2. This would be the factored form if we use factoring by grouping. But notice that this doesn't help us out. None of these factors appears in our denominator. We can't simply any more. I final answer would be 6 x squared plus 5 x plus 1. Divided by the factors of this denominator.
Try this third subtraction problem now. Be sure you distribute this negative sign to each term in our numerator. Then, you'll have an addition problem and you can proceed as usual. Good luck. You'll also want to get the most simplified answer. Multiply out your numerator and then leave your denominator in factored form.
Here's our answer. Great algebra skills if you found that one out. First we distribute our negative sign to each term in our numerator. So we'll have plus negative two b plus five. Make sure you change that sign there. Next we find the lowest common denominator, which is b plus two, times b minus two, times b plus five. Next we multiply each fraction by a form of one. This denominator was missing b plus 5, and this denominator was missing b plus 2. Next we multiply these two numerators together, to get b squared plus 8b plus 15. For the second equivalent fraction, we'll multiply negative 2b plus 5, times b plus 2. This gives us this trinomial, and finally we add the like terms of each numerator together. B squared and negative 2b squared equals negative 1b squared. 8b and positive b equal 9b and 15 and 10 equal 25. So this is our final answer. We also know that this is our final answer because we can't factor this enumerator further. There are no factors of negative 25. That's sum to positive nine. I hope you've gained a lot of practice with adding and subtracting rational expressions.