WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at adding and subtracting rational expressions.
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In order for this process to workout correctly I have to spend a little bit of time working on finding a least common denominator.
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This will help us so we can rewrite our rational expressions and actually get them together.
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Then we will get into the addition process.
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We will look at the some examples that have exactly the same denominator.
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We will look at others which have different denominators.
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Once we know more about the addition process, the subtraction process will not be too bad.
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You will see that it will involve many of the similar things.
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So far we have covered a lot about multiplying and dividing rational expressions,
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but we also need to pick up how we can add and subtract them.
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One key that will help us with this is you want to think of how you add and subtract simple rational number.
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Think of those fractions, how do you add and subtract fractions?
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One key component is that we need a common denominator before we can ever put those fractions together.
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Think of how that will play a part with your rational expressions.
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Let us first look at some numbers okay.
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Let us suppose I had 2/1173 and 5/782 and I was trying to add these or subtract them.
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No matter what I'm trying to do, I have to get a common denominator.
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One thing that will make this process very difficult is that I chose some very large numbers in order to get that common denominator.
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One thing that could help out when searching for this common denominator is not to take the numbers directly since they are so large.
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You have to break them down into their individual factors.
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Down here I have the same exact numbers, but I have broken them down into 3 × 17 × 23.
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I have broken down the 782 into 2 × 17 × 23.
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What this highlights is that the numbers may not be that different after all.
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After all, they both have 17 and 23 in common and the only thing that is different is this one has a 3 and this one has a 2.
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When trying to find that common denominator, I want to make sure it has all the pieces necessary
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or I could have built either one of these denominators.
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It must have the 2, 3 and also those common pieces of the 17 and the 23.
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This is the number that you would have been interested in for getting these guys together.
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When working with the rational expressions, we will be doing the same process.
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We want them to have exactly the same denominator, but I would not be entirely obvious to look at the denominator and know what that is.
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We will have to first factor those denominators, we can see that the pieces present in which ones are common and which ones are not.
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We will go ahead and list out the different factors in the denominator first.
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We will also list out the variables, when it appears the greatest number of times.
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We will list the factors multiplied together to form what is known as the least common denominator.
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Think of the least common denominator as having all the factors we need and we could have built either one of those original denominators.
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Let us take a quick look at how this works with some actual rational expressions.
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I want to first look at factoring the bottoms of each of these.
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8 is the same as 2 × 2 × 2 and y⁴ we will have bunch of y all multiplied by each other.
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For 12 that would be 3 × 2 × 2 and then a bunch of y, 6 of them.
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When building the least common denominator, I will first gather up the pieces that are not the same.
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Here I have the 3 and this one has an extra 2.
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I need both of these in my least common denominator 2 and 3.
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This one has a couple of extra y.
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Let us put those in there as well.
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Once we have spotted all the differences between the two then we can go ahead and highlight everything that is the same.
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We have a couple more twos and 1, 2, 3, 4 y.
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Let us package this altogether.
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2 × 2 × 2 = 8 × 3 = 24 and then I have 6y, y⁶.
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Let us do some shortcuts here.
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One, you could have just figured out the least common denominator of the 8 and the 12 that will help you get the 24.
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You can take the greatest value of y⁶ and gotten the y⁶.
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We use techniques like that to help us out when looking for that least common denominator.
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Some of our expressions may get a little bit more complicated than single monomials on the bottom.
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Let us see how this one would work.
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This is 6 /x² - 4x and 3x – 1/ x² – 6.
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In order to figure out what our common denominator needs to be, we are going to have to factor first.
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Let us start with that one on the left and see if we can factor the bottom.
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It looks like it has a common x in there.
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We will take out and x from both of the parts.
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For the other rational expression that looks like the difference of squares.
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x + 4 and x – 4
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We just have to look with these individual factors.
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I can see that what is different is this x + 4 piece and the x piece.
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Let us put both of those into our common denominator first.
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I have x and x + 4 then we can go ahead and include the pieces that are common.
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Any common pieces will only include once.
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This down here represents what our least common denominator would be.
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We need an x, x + 4, and x -4.
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Finding the least common denominator is only half the battle.
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Once you find the least common denominator, you have to change both of your rational expressions
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so that they contain this least common denominator.
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Once you have identified it go one step farther and rewrite the expressions so that they have this least common denominator.
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Let us watch how this works with our numbers.
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That way we could get a better understanding before we get into the rational stuff.
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Here are these fractions that I had earlier and you will notice that the bottom is already factored.
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Our LCD in this case was 2 × 3 × 17 × 23.
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Now suppose I want them to both have this as their new denominator.
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2 × 3 × 17 × 23, 2 × 3 × 17 × 23.
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When looking at the fraction on the left here the only difference between this and the new LCD that I wanted to have is it is missing a 2.
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I could give it a 2 on the bottom but just to balance things out I will also have to give it a 2 on the top.
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On the top of this one will be 2 × 2.
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I better highlight that this 2 was the one we put in there.
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For the other one, it needs to have that 3, I will give it a 3 on the top and there is where the 3 came from in the bottom.
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You can see that we give the missing pieces to each of the other fractions.
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If I was looking to add or subtract these I will be in pretty good shape since they have exactly the same denominator.
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We want to do the same process with our rational expressions, give to the other rational expressions its missing pieces
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so it can have that least common denominator.
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We can find a least common denominator now, which means we can get to the process of adding and subtracting our rational expressions.
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Think of how this works with our fractions.
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If I have two fractions and I have exactly the same denominator then I will leave that denominator exactly the same
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and I will only add the tops together.
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This works as long as my bottom is not 0.
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This will be the exact same thing that we will do for our rational expressions, the only difference is that this P, Q, and R
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that you see is my nice little example, all of those represent polynomials instead of individual numbers.
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As soon as we get our common denominator we will just add the tops together.
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If they do not already have a common denominator, we have to do a little bit of work.
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It means we want to find a common denominator and often times we will have to factor first before we can identify what that is.
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Then we will have to rewrite the expression so that both of them have this least common denominator.
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Once we have that then we will go ahead and add the numerators together and leave that common denominator in the bottom.
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Even after that we are not necessarily done.
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Always factor at the very end to make sure that you are in the lowest terms.
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Sometimes when we put these together we can do some additional canceling and make it even simpler.
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The subtraction process is similar to the addition process.
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You will go through the process of finding the common denominator.
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Make sure they both have it, and then you will end up just subtracting the tops.
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Remember though you want to subtract away the entire top of the second fraction.
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Often to do this, it is a good idea to use parentheses and distribute through by your negative sign on the top part.
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That way we would not forget any of your signs.
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It is usually a very common mistake when subtracting these rational expressions.
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Let us get to business.
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Let us look at this example and add the rational expressions.
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I have (3x / x² – 1) + (3 / x² – 1).
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The good news is our denominators are already exactly the same.
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I will simply keep that as my common denominator in the bottom and we will just add the tops.
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Even though we have added this and put into a single rational expression, we are not done.
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We want to make sure that it is in lowest terms.
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Let us go ahead and factor the top and bottom, see if there are any extra factors hiding in there.
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As I factor the top, this will factor into 3 × x +1 and then we can factor the bottom, this will be x + 1 and x – 1.
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I can definitely say yes there is a common piece in there, it is an x +1 and we can go ahead and cancel that out.
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I'm left with a 3 / x - 1 and now I have not only added the rational expressions, this is definitely in lowest terms.
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Let us try this on another one.
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We want to add together the two rational expressions I have (-2/w + 1) + (4w/w² -1).
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This one is a little bit different.
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The denominators are not the same.
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Let us see if we can figure out what the denominator should have in the bottom by factoring them out and seeing what pieces they have.
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w² - 1 is the difference of squares which would break down into w + 1 and w -1.
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It looks like that first fraction is missing a w -1.
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We have to give it that missing piece.
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We will write it in blue.
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I will give it an extra w -1 on the bottom and on the top just to make sure it stays the same.
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Our second fraction already has our least common denominator, so no need to change that one.
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Now that it has a common denominator we will keep it on the bottom and we will simply add the numerators together.
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I got -2w -1 + 4w.
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We cannot necessarily leave it like that, I’m going to go ahead and continue combining the top,
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maybe factor and see if there is anything else I can get rid of.
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Let us distribute through by this -2.
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-2w + 2 + 4w / (w +1) (w -1)
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(-2w + 4w) (2w + 2)
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I think I already see something that will be able to cancel out.
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Let us factor out a 2 in the top.
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Sure enough, we have a w + 1 in the top and bottom that we can get rid of.
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That is gone.
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Our final expression here is 2 / w -1 and now we have added the two together and brought it down to lowest terms.
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Let us do a little bit of subtraction.
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This one involves (5u /u – 1 – 5) + (u /u -1).
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This is one of our nice examples and that we are starting off and has exactly the same denominator.
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That is good so we can go ahead and just subtract the tops.
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I will have 5u - the other top 5 + u.
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Note what I did there, I still have the entire second top and I put it inside parentheses and I'm subtracting right here.
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One common mistake is not to put those parentheses in there and you will only end up subtracting the 5.
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You do not want to do that.
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You want to subtract away the entire second top.
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To continue on, I want to see if there is anything that might cancel.
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I’m going to try and crunch together the top a little bit and see if I can factor.
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Let us distribute through by that negative sign.
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5u - 5 - u / u -1
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That will give me 5u – 1 = 4u – 5/ u – 1.
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It looks like the top does not factor anymore.
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This guy is in lowest terms.
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Let us tackle one more last one and these ones involve denominators that are not the same.
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We are going to have to do a lot of work on factoring and seeing what pieces they have before we even get into the subtraction process.
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Let us go ahead and factor the rational expression on the left.
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On the bottom I can see that there is an (a) in common.
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Over on the other side I can factor that into a - 5 and another a – 5.
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They almost have the same denominator.
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They both have that common a - 5 piece and the one on the left has an extra a.
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And one on the right has an additional a – 5.
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Let us give to the other one the missing pieces.
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Here is our rational expression on the left, we will give it an additional a – 5.
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With this one it already has a - 5 twice, we will give it an additional a on the bottom and on the top.
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Now that they have exactly the same denominator we can focus on the tops 3a a– 5 - 4a /(a – 5) (a- 5).
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It looks like we can do just a little bit of combining on the top.
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I have (3a² - 15a – 4a) / (a)(a – 5)(a-5)
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When combined together the -15a and the -4a, 3a² – 19a.
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We have completely subtracted these we just need to worry about factoring and canceling out any extra terms.
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On the top I can see that they both have an extra a in common, 3a-19.
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Let us take that out of the top.
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We will cancel out that a and now we brought it down into lowest terms.
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I have 3a - 19 / (a – 5) (a -5)
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Whether you are adding or subtracting rational expressions make sure that you have your common denominator first.
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Once you do, you just have to focus on the tops of those rational expressions by putting them together.
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Once you do get them together remember you are not done yet, feel free to factor one more time and reduce it to lowest terms.
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