Now that we've seen how to simplify rational expressions, let's use that knowledge to help us multiply and divide them. But before we can multiply and divide rational expressions, let's think back how to multiply fractions. What would you get if you multiplied these two fractions together? Write your answer here.
Great work if you got 6 divided by 55. When we multiply fractions, we multiply the numerators straight across and the denominators straight across. And if you recall all the way back to our first unit, we want to factor the numerators and denominators first to simplify our math. We know 4 divided by 4 equals 1, and 3 divided by 3 also equals 1. Keep in mind I can only simplify one of these 3's, since one is the numerator and one is the denominator. This other 3 must stay in the numerator since there isn't another 3 in the denominator to cancel it out or simplify to 1. We're left with 2 times 3, which is 6, and 11 times 5 which is
We can use the same approach when we multiply rational expressions. So, even something that looks really complicated like this, we can simplify to a smaller fraction. First, we want to factor the numerators, and then factor the denominators. Once we've done that step, we want to cancel or simplify the common factors that appear in the numerator and denominator, since they'll equal get? Put it here.
x plus 4 divided by x plus 1 is correct. For the first numerator, we find the factors of 12 that sum to positive 7, which are positive 3 and positive 4. For the second numerator, we find the factors of positive 4 that add to negative 5. These are negative 4 and negative 1. Notice that the a coefficient in front of the x squared term is 1 in every single quadratic. This means that we can just use the factors we find of 12 That summed to 7 in our binomials. The same was true here, we used the factors of positive four that sum to negative five and four in our binomials. These are the factors for our other denominators and now we're ready to simplify. X plus 3 divided by x plus 3 equals 1. And x minus 4 divided by x minus 4 Also equals 1. The last equivalence of 1 is x minus 1 and x minus 1. Remember that these factors don't dis, disappear, they really equal 1. So we multiply 1 times x plus 4 and one's times x plus 1. Any times anything is itself. So we're left with x plus 4 divided by the quantity x plus 1.
Let's try another practice problem. What do you think would be the result of multiplying these two fractions together? Write your answer here.
You might not be sure what to cancel here. So let's look at one fraction, and our factors multiplied together. First I'm going to expand this power of two. The quantity x plus y squared is really x plus y times x plus y. Now we just multiply our numerators across and our denominators across. I need to make sure to put parentheses around this x plus y, since we would really be distributing this 2x to each of the terms. We could see that here. But I don't actually want to distribute or multiply the 2x to each term, since you want to be able to cancel factors here and here. These factors reduce to 1, and we're left with a 1 in the numerator and an x plus y in the denominator. You want to make sure that you don't think the numerator is zero. These factors really reduce to 1, so we have 1 times 1, which is 1. Some fractions will end up with a 1 in numerator, and others might have a 1 in the denominator. You'll want to watch out for that.
Try multiplying these two fractions together. You'll want to factor the numerators and denominators first, and then, cancel any factors that reduce to one. Write your answer here and leave it in factored form. For example, if your fraction multiplies to this, you don't want to multiply your two factors together in the numerator. Just leave them factored.
A plus 3 times a plus 5 is correct, great work if you figured that one out. We can start by factoring the two numerators. The factors of negative 5 that sum to factors of 15 that sum to 8, those are positive 5 and positive 3. We know we're going to multiply these two fractions together, so we'll multiply every factor in the numerator and every factor in the denominator. We can think of this denominator as 1 times the quantity a plus 5. The same is true for this one, 1 times a minus 1. I show this so that way we can understand we 're really multiplying these two factors together. This is why I use the parentheses around these two expressions. We know a plus 5 divided by a plus 5 equals 1 and a minus itself so this is our answer.
Now, we're going to try something a little more complicated. For this problem, you'll want to start by factoring each parts of the fraction. We'll start there first and check in and then we'll cancel factors after that.
These are the correct factors for each part of the fractions. I know factoring can be tough, so if you're getting at least half of these right, great work. If you got stuck on one of these, try to look back at your work and see where you went wrong to get these factors. If you can't find an error in your work then stay with me for this solution. For our first numerator we can pull out a 2x from each term. So we'll have 2x times x squared minus 6x plus 9. Then we notice that this is a special factoring pattern. It's a perfect square trinomial. So we have 2x times x minus 3, times x minus 3. This gives us our first numerator. For this denominator, we want to find the factors of negative 12, that sum to negative 4. This allows us to rewrite our middle term. And then we'll use factoring by grouping to get 3x plus 2 times x minus 2. This is our factored form for our first denominator. For this numerator we want to find the factors of 12 that sum to negative 8. These two factors are negative 6, 6 and negative numerator. And finally for this last denominator we pull out a 6x from both terms leaving us with 6x times x squared minus 9. Then we can factor this difference of squares using x plus 3 times x minus 3. This difference of squares pattern appears in all sorts of math. So it's great that we can recognize it quickly.
Here are our fractions in factored form. Now, I want you to use your knowledge of canceling factors to simplify this multiplication problem and then write your final answer here.
We can cancel an x minus 3 in the numerator and the denominator. We can also cancel an x minus 2. Now we're left with these factors. We want to be careful here. We can't simplify 3 x minus 2 and 3 x plus 2. They're not identical rational expressions. And they're not opposite polynomials, since the 3 x have the same signs. The same is true of x minus 3 and x plus 3. The signs for the x terms are the same. So these are not identical polynomials, and they're not opposite polynomials. We can, however, simplify 2x and 6x. You know, x divided by x equals 1. And 2 divided by 6 reduces to 1 third. You want to make sure you don't forget about monomial terms. Be sure to cancel common factors for the numbers. And common powers for the variables. After we've done that we have our simplified rational expression, 3x minus 2 times x minus 3 and 3x plus 2 times 3 times x plus 3.
Let's try another problem multipying rational expressions. Again, we're going to break this problem into two parts, first I want you to factor each part of the fractions and then we're going to cancel the factors that appear and the numerator and denominators in the second part. You can write the factor expressions here in the corresponding boxes. Be sure you use all that you know about factoring. Look for a greatest common factor first and then use your knowledge of factoring patterns and factoring by grouping if necessary.
These are the factored forms. Great work for getting at least half of these right. If you really struggled with these, I would go back and review factoring. Try and get in some more practice before moving on. If you only missed one or two, then see if you could find your mistakes by looking at your factors. If you can't find your mistakes, then continue watching the solution to see where your reasoning went wrong. The greatest common factor for these three terms is one. So let's use factoring by grouping to get our two factors. We want factors of negative 30 that sum to negative 1. The factors are negative want to find factors of negative 20, that's sum deposit of 8. The factors would be positive 10 and negative 2. We use factoring by grouping to get our denominator of 2x plus 5 times 2x minus 1, here. Next, we factor this trinomial. The greatest common factor for each term is one, so we need to find factors of negative 60 that sum to negative 7. We don't need to factor out a g c f. These two factors are negative 12 and positive 5. They might have been tough to find. We use factoring by grouping to get 2 x plus 5 times x minus 6. And this is what we put in our numerator. For our second denominator, we want to find factors of negative 30 that sum to positive 13. We use factors by grouping to get our factors of x plus 3 and 5x minus 2. And now we're done factoring, and we're ready to simplify.
Use your knowledge of opposite polynomials and canceling factors to simplify this multiplication problem. Write your answer here. And here again, leave your answer in factored form. Don't multiply out any factors together.
Here are the common factors that simplify to 1. This leaves us with 3x plus 1 times x minus 6 in our numerator, and x plus 3 times 2x minus 1 in our denominator. Nice thinking if you got here.
Here's another multiplication problem. Try finding the factored form for each part of the fractions and write your answers here.
Here the factored forms for each part of the fraction. Great work if you got all of these correct. If you made a mistake that's okay too. Now's a great time to go back and look over your work. See if you can figure out how to get these factors. If not, you can always watch the solution. For our first numerator we can remove a 2 x squared as the greatest common factor. This leaves us with x squared plus 4 x plus 3 for our second factor. Now we can easily factor this one by finding factors of 3. That sum to positive 4. That would be positive 3 and positive 1. This is why it's important to find a greatest common factor first. If we don't find this greatest common factor we won't be able to factor each these and cancel our factors in the end. For this denominator we have a coefficient of 1 in front of the x squared term. So we just want to find factors of negative 15 that add to 2. This would be positive 5 and negative 3. For this numerator the greatest common factor of the three terms is 1. So we just want to find factors of negative 50 that sum to positive 5. The factors would be negative 5 and positive 10 and then we can use factoring by grouping. To get our factors of x plus 5 and 2 x minus 5. For our last denominator we can remove a common factor of 6 x from every single term. If we factor the 6 x as a greatest common factor, we're left with 2 x squared plus 3 x plus 1. If we distribute this really quickly we can make sure that this is correct. When we look at factoring this expression, we want to value in factors of positive 2 that sum to positive 3. We can use factoring by grouping. To arrive at the factors of x plus Now we're ready to simply our fractions since everything's in factored form. Great work.
Alright, you've done all the legwork to get up to this point. Now reap the rewards and simply cancel the factors in the numerator and denominator. Write your final answer here. And again, leave your answer in factored form. Don't multiply it out like these. Leave it like this.
This is the simplified fraction, fantastic work if you found it. I think it's best to start by reducing the factors of x plus one and x plus five. These factors in the numerator and denominator cancel or reduce to one. I don't want to forget about these monomial terms, but everything looks a little confusing right now. So I'm going to rewrite my line. So we have two x-squared, times x plus three, times two x minus five, for our numerator. For the denominator we have x minus 3 times 6x times 2x plus 1. Notice that I switched the order of this 6x and this x minus 3. We can do that since we're multiplying here. We know the multiplication is commutative. We know 2/6 reduces to 1/3 so I can simplify this fraction. We can also simplify our powers of x we have x squared in the numerator and 1 x in the denominator so we'll be left with 1 x in our numerator in our final answer. I'll show it more easily by writing x squared as x times x. We simplify each of these x's since they really equal 1, and 2 divided by 2 equals 1, and 2 into 6 equals 3. This gives us our final answer.
Now that you've gotten some practice with multiplying rational expressions, let's move on to dividing rational expressions. And instead of diving right into dividing rational expressions, let's start with what we know, dividing fractions. If we want to divide by a fraction, we really multiply it by the reciprocal of that fraction. So three-sevenths divided by two-sevenths would be three-sevenths times seven-halves. We cancel the factors in the numerator, denominator which leaves us with three halves. We divide rational expressions the same way that we divide fractions. We're going to change division to multiplication and multiply by the reciprocal. So try doing this rational expression problem. Change the division to multiplication and multiply by the reciprocal of this fraction. Make sure you find the cube of 2t before you multiply. When you think you've got it, write your answer as a fraction here.
First we cube the quantity 2t. The quantity 2t cubed is 8t cubed. Now that our fraction is simplified, we can change division to multiplication, and multiply by the reciprocal of the second fraction. We just flip this fraction over. I'm going to multiply these fractions straight across to get negative 24t to the the common factors here. Which is great. We know 8 divides into 24 three times and 8 divides into 40, 5 times. When I look at my t's, I notice that I'll have four more t's in the denominator, so I'll have negative three divided by five times t to the fourth. That's some amazing algebra if you got that one correct.
lets try another division problem. We are going to divide this rational expression by this rational expression. Remember to change your division to multiplication and then multiply by the reciprocal of the second fraction. Then your problem turns into something that you've definitely seen before. You can write your final answer here
p plus 4 divided by the quantity p plus 2 is correct. Impressive work if you've got that correct. To divide by a rational expression, we multiply by the reciprocal of the second fraction. Now, we really just have a multiplication problem that we're familiar with. We factor this trinomial to get p plus 5 time p minus 1 and we factor this denominator to get p plus 5 times p plus 2. These factors reduce to 1 and we're left with p plus 4 in our numerator, and p plus 2 in our denominator.
Try dividing this rational expression. Write your answer here.
Fantastic work if you've got 12 n divided by 5. This 3 n minus 6 might have given you trouble so let's see how to do it. We changed division to multiplication, and multiplied by the reciprocal of our second fraction. Also you might have noticed that I factored a 3 from the first two terms in the numerator. I factored this and since I knew I was going to be multiplying my numerators and my denominators together. I want to be able to find factors that reduce to 1. You don't need to show all your simplification at once. So a lot of times I think it's best to rewrite what's remaining. When we write out all the factors, we can see that the 5s would reduce to 1 and one n would reduce to 1, leaving us with 12 n divided by 5.
Try dividing these two rational expressions. Write your final answer here.
Here's our factored answer. Great work if you got that one correct. To divide a rational expression, we multiply by the recripricol. So, we change division to multiplication and flip the second fraction over. We can use factoring patterns to find the factors of this numerator and. And denominator. This is a difference of squares. So we have x plus 3 times x minus 3. Our denominator, however, is a perfect square trinomial. So we'll have x minus 3 times x minus 3. We find factors of negative 6 that sum to negative 5. Those factors are negative 6 and positive 1. We use factoring by grouping to get 2x plus 1 times x minus 3. For our denominator, we want to find factors of negative 36. That sum to negative 5. Those factors are negative 9 and positive 4. After factoring, we get 3x plus 2, times, 2x minus 3. Now we're ready to simplify our factors. This x minus 3 divided by x minus 3 equals 1. And look at that, there's another x minus 3, divided by x minus Is 3. Those will also equal 1. And our remaining factors are x plus 3 and 2x plus 1 for the numerator, and 3x plus 2 times 2x minus 3 in our denominator.
Try dividing these rational expressions. Write your final answer here.
R plus 6 s divided by r plus s is correct. That's some amazing work with rationals if you're getting it right. You might have been stumped on how to factor one of these. If so, watch me do the first one, and then see if you can finish tackling this problem. To divide by any fraction, including a rational expression, we change division to multiplication and multiply by their reciprocal of the first fraction. Now we want to start factoring each piece of these fractions. I'm going to start with this first one. We know the first two terms must be r since r times r is r squared. Now we're just trying to find factors of negative 12 that sum to positive 1. Since there's a 1 in front of this r times s. We know that positive 4 and negative 3 multiplied together is negative 12, and sum to give us positive 1, but notice that I need a negative 12 s squared for my last term. This must mean that I need a variable of s after each of these numbers. We can quickly check this factoring by multiplying out the binomials in our head. R times r is r squared, and 4s times negative 3s is negative 12s squared. This positive 1 r s comes from multiplying these two terms together, and adding it to the result of multiplying these two together. So we have negative 3 r s plus 4 r s, which is 1 r s. For this first denominator, we want to find factors of negative 20 that sum to negative 1. For our second fraction we want to find factors of negative 30 that sum to 1. That would be positive 6 and negative 5. And then for our denominator of our second fraction we want to find factors of negative 3 that sum to negative 2 which are negative numerator and denominator. There, r minus 3 s divided by r minus 3 s equals 1. And r plus 4 s divided by r plus 4 s also equals 1. The last factor we need to cancel is r minus 5 s. And this leaves us with our final answer, r plus 6 s divided by r plus 1 s, or just s.
For our next problem, I want you to find the quotient or the answer, to this division problem. Write your answer in factored form here.
As always, we start by changing division to multiplication, we multiply by the reciprocal of the second fraction. I'm multiplying these binomials together, so I want to make sure that I use parentheses. I have x plus 5 and 5 plus x. If I think back, I can remember that these two are actually identical, they're the same factor. So, we can simplify those or cancel those to make one. This leaves me with 5 minus x in my numerator and x minus 5 in my denominator. And here we have opposite polynomials. The 5's are opposite in sign as are the x's. We know opposite polynomials really reduce to negative 1. So negative 1 is our final answer.
Try dividing these rational expressions. Write your final answer here in factored form.
We start our problem by multiplying by the reciprocal of our second fraction. We flip it over. We can factor this first numerator as 5x plus 3 times 5x minus 3, since this is the difference of two perfect squares. Our denominator is also a difference of two perfect squares. So we have 2x plus 3 times 2x minus 3. For this fraction, we can factor this numerator by finding the factors of negative factoring by grouping to get 3x plus 1 times 2x minus 3 for this first numerator. And finally, for this denominator we find factors of positive 45 that sum to negative 14. Those two factors are negative 9 and negative 5. These factors let us rewrite our middle term, and then we can use factoring by grouping. We get a denominator of 5x minus 3 times 3x minus 1. Now that we have all of our factors, let's cancel factors that simplify to 1. These two factors would reduce to 1, and these two factors would also reduce to 1, leaving us with these factors for our answer. You're doing amazing at dividing rational expressions and factoring if you got this one correct.
All right, For our 7th and final problem I want you to try dividing these two rational expressions. This is a challenge problem, so it's going to be a little bit tougher than one's you're used to. I'll admit, the hardest part is actually factoring, but if you do get the correct answer then great job. You're on your way to becoming a pro at factorizing and rationalizing expressions.
Okay, if you attempted to solve this, you should have gotten these as your factored fractions. Now there's a lot of missing work in between. You needed to change division to multiplication and take the reciprocal of that second fraction. I'm not going to show you all the factoring for each of these, but hopefully you can figure it out. In the end, we can simplify the factors a plus divided by 3a minus 4. You want to be really careful at the end here, as well. These last two factors have different coefficients for the a. One is 4, and one is 3. You might think that these are opposite polynomials, but we know that's not true since the a's have different coefficients, and the constants are different as well. We can't simplify these at all, so we'll leave this as our answer.