Add Rational Expressions‬‏_2 - Robichaud.mp4

Quiz 4, number 5. Here we have three algebraic fractions or rational expressions being added together. So just like in arithmetic, when you have three fractions being added together without any variables in them, we want to use the same process. We want to find common denominators. Then we want to make each fraction half the common denominator as its denominator. And from there, you add across the numerators, keep the common denominator, and make one single fraction. And we're going to also want to try to do any canceling out we can here. So the first step we want to do is factor the denominators. Because once we factor the denominators, we're going to be able to see what the least common denominator is much easier. The first fraction cannot be factored, the second one, no factoring needed. Third one, we can pull out the greatest common factor of y. Now that we have our factored denominators, we can build our least common denominator, the LCD. Take the first denominator that you see entirely. Now remember, y plus 6 in its entirety is one factor. And factors are things that are going to be multiplied together. The y and the 6 here are not factors of each other because they are being added together. Now you look at the second denominator. And if there any factors that are in the second denominator that are not contained already any LCD, you need to multiply them in. So the 3 is being multiplied. It's a factor. I need to put in because I didn't see it in the y plus 6. And the y is a separate factor also. Then you look at your third denominator and see if you already have all the factors contained in the LCD. y-- yes, a separate factor of y is here. y plus 6-- yes, we already have a separate factor of y plus 6. So that's our entire LCD. We have each denominator contained in here entirely with no extra pieces. So rewriting the LCD, we get 3y times y plus 6. Now what we need to do is rewrite each fraction so that it has the common denominator as its denominator. So we need to look at each fraction and see what's missing in the denominator. y plus 6-- well, we need to multiply it by 3y if we want to turn that into the LCD. So let's multiply by 3y. And that's OK to do as long as we do the same to the top. 3y-- it's missing the y plus 6. So we're going to multiply y plus 6, top and bottom. And the last one, y, y plus 6 are already contained. We need a 3. So we're going to do times 3, top and bottom. So let's finish the rewrite. Let me first fraction. We have the numerator 6y, because it's 2 times 3y over the common denominator of 3y y plus 6 plus. Second fraction, we have 5 times y plus 6 over the common denominator 3y y plus 6. Third one, we have 12 times 3 or 36 over common denominator 3y times y plus 6. Now we all have the same denominator. So that means we're going to be able to add the fractions. So what I'll call this is the make one fraction step. We're going to keep the common denominator and just add across numerators. Now this one right here-- 5y plus 6-- once we distribute that 5, it's going to be 5y plus 30, and then plus 36 for the last piece. Now we want to simplify, which means in this case, collect like terms in the numerator. So we do 6y and 5y is 11y plus 30 and 36 will be 66, all over the same denominator. And now that we've simplified, the next step we want to do is see if we can cancel anything out. But remember, as far as canceling out, always factor before canceling. All right, so you don't want to go in here and say, well, this 3 cancels out with the 66, because 66 divided by 3 gives us 22. You don't want to cancel like that. You've got to factor first, because you can only cancel common factors. So we need another factor step. The numerator factors into 11 times y plus 6. And keep the denominator. Now we look for common factors-- cancel common factors. All right, so it has to be factors. y plus 6-- yes, that's a common factor being multiplied into the top and bottom. It's gone, and nothing else cancels. So now we're going to have 11 over 3y. And that's our final answer.