## Transcripts

1. 4.1 Operations with Algebraic Fractions: Lowest Common Denominator : Hello, everyone. Welcome to the adding and subtracting algebraic fractions video In this video, we're going to go over how to add and subtract. Um, I was a break fractions and also had a multiply and divide as well. And if you need some refresher on polynomial, I have ah few courses. Um, in the beginning before this that have the operation with polynomial is which will allow you to understand this course better now first for adding and subtracting algebraic fractions. Let's write down some rules. So first we have to find the LCD, which is a lowest common denominator. Um, then we place all non fractions over one, and then we factor all the denominators. So let's start with the problem here. So this is the equation. It's five x over X plus three times X minus three plus two over x r eight X squared plus X o X plus four over X minus three. Now what we're gonna be looking for is the, um, lowest common denominator We do is we Look at the terms we have eight x square, and then here we have two common ones of X minus three. So it's eight x squared times X minus three and then we can take this one X plus three. So now we can take these terms and what we want to do is we want to convert these each of these terms or each of these denominations into this denominator in the best way to do that . And what I'll do is I'll write this over here. What we'll do is we'll multiply in order in order to get this denominator to that, we have to multiply by eight X. But of course, you have to multiply both the numerator and the denominator. Otherwise, you change the fraction because of you. Multiply this by eight x squared and we'll buy this by eight x squared. What you're really doing is you're multiplying it by eight x squared over eight x squared, which is one. So it doesn't matter what those numbers are, the fraction will not change. So what we do is we multiply that one. So we end up with we end up with 40 x cubed over eight X squared, X plus three X minus three. Now we can then take this one. Now this one already has the eight x squared but it doesn't have the X plus three or X minus three. So what we can do is we can multiply this by x minus three x plus three by this by X minus three X plus three and what we get is to x minus three x plus three over eight X square, X minus three X plus three and then we can take this one here. Now. This one has the X minus three, but it doesn't have the X or the eight X squared or the X plus three so we can multiply this by eight x squared X plus three A X square X plus three and it becomes so this becomes X plus three or X plus four times X minus or plus three times eight x squared over eight X squared X minus three X plus three So that is how to find the lowest common denominator in a fraction. Now, the next video, we're actually gonna take this completed fraction, and now we're going to simplify it or we're gonna actually added together and in solve it
2. 4.2 Operations with Algebraic Fractions: Adding Algebraic Fractions: So in the last lesson, we, as you can see here we went from five x over X plus three x minus three plus two over X squared eight X square plus X plus four over X minus three. We took that and we found the lowest common denominator. Which lows coming in The denominator was at eight X was eight x x plus three three X minus three. We took that and we multiplied it through and way. Got 40 x to the third over eight x squared x plus three x minus three plus two x minus three x plus three over eight X squared x minus three X plus three plus x plus four X plus three eight X squared over eight X squared X plus three X minus three. Now we're going to take this and we're actually gonna add it all together. Now that it's been, um, the lowest common denominator has been found. So now that we have this equation, we're able to Then now that they're all lowest common denominator, now we can solve this. Now what? As I as I showed with my previous videos in this case, now that we have this, we can actually, as a lowest common nominator weaken, treat this as combining like terms of combining polynomial is adding and subtracting polynomial. So now what we want to do first is we want to get rid of all these, um, correct disease. Now, here there's no parentheses in it, is what it is. So we can write 40 X to the third, and what I'm gonna do is I'm just gonna treat handle this. I'm not even going to deal with the denominator yet for us, and then we can take this one and multiply it through 1st 1st we can do is we can handle the parentheses, so x Times X is X squared, and then x times three is three X and then negative three times X is negative three x and then X and then three negative three times are the times negative or negative? Three times three is nine is negative nine. And then this is all multiplied by two, which we can Then further, uh, put this down to two X. We can distribute the two throughout this polynomial two x squared plus, and in these two can actually cancel out because they're plus three x minus three x and then you have nine times to with it which is are negative nine times to which is negative. 18. So then we can go through this one, which I like again. I like to handle the parentheses first. So X squared in an X Times three is three X and then four times X is four X and then four times three is 12 and then we can take this and distribute the eight x squared through it, which here we get we get, um, e X to the fourth, and then we have plus 24 x huge plus 32 X cube plus 96 x squared. So now we have this big, messy long polynomial which we can then find the light terms, which right now, you start with the highest term, um, exponents downwards, which right here we have, uh, to the fore. We have only one of them. So we'll right that here eight x to the fourth. And I like again, I like to cross things out. Um, it exited fourth. And then now we have our cubes, which we have, um 40 24 and 32 which 40 plus 24 plus 32 96. So we have plus 96 x Q and then so that takes care of the cubes. Then we have the squares, which is 96 plus two. So it's 98 plus 98 x cubed r squared, that takes care of those. And then finally, we have the minus 18 and now we can take that and place it over E X squared X plus three or X minus three times X plus three. And now, of course, depending on how you do this, you can decide to go further and simplify it. Um, find common factors. Like, uh, we could easily find, um, two as a common factor, because these are all even numbers, um, all that kind of stuff. But for right now, that's it for adding algebraic fractions.
3. 4.2 Operations with Algebraic Fractions: Multiplying and Dividing Algebraic Fractions: So in the last video, we did so in the last lesson we did adding and subtracting algebraic fractions. Now we will be doing multiplying in dividing complex fractions. Now, as we know with, um, regular fractions when you multiply and divide. If you have, let's say to over seven times 8/6, we don't have to worry about finding a lowest common denominator. All we have to do is multiply through. We just multiply both of them. Now, if we divide now, when we divide, what we do is we actually multiplied by the reciprocal. So in this case, this would then reciprocate turn into eight. The over 16 divided by 1/2, would change to 16 to 8/16 times 2/1, which eight times two is 16 in a 16/1 of 16. So that would equal one now this into its and actually translates when you, when you are, speak in terms of algebraic fractions. So let's say we take two x squared plus six x over X squared minus nine times three minus X over five x. Well, in this case, um, if you took my course on, um multiplying and dividing polynomial. If you took my course on multiplying polynomial, it's pretty much you're taking to separate. Multiplication is and dividing them. It's pretty easy. So what you do is what you do is you could kind of go foil it through. You have two X squared times three at six x squared. Then you have two X squared times X, which is two X. You then you have six X times through actually my sex so you have it becomes minus and then you have six X times three, which is 18 X and then you have six x squared minus X times minus X, which is minus six X squared, and then you go through on the bottom. You have X squared times five x, which is X R five x cubed and then you have negative nine x Times five, which is 40 negative 45 x and pretty much it's just a matter of multiplying it through is actually pretty easy. Now let's flip things around and do division again. Same intuition as with non algebraic division of fractions. Um, all you have to do is multiplied by the reciprocal, so this one stays the same two x squared plus six x over X squared minus nine. But then this time this one becomes a reciprocal in multiplication. So you multiply by five x three minus x and then we just multiple are multiply through You have to x times five x two X Squared times five x, which is 10 x You plus six x times five x, which is 30 x squared over in an X Square Times three, which is three x squared. An X squared times X, which is negative x, which is negative. X Cube. You gotta watch for that negative and then negative nine times three, which is negative. 27 in a negative nine. Next times negative X, which makes a positive nine x and that is how you divide the complex are algebraic fractions. Now one thing you might run into is complex fractions in this form, which is pretty much the same thing. All you do is you take this section here and you flip it around two reciprocal, and then you multiply this or this by this, and you get the same result as last time. So that's the course on adding, subtracting, multiplying and dividing outbreak fractions. Um, thank you for checking out my videos. Check out my past videos if you haven't and, uh, wait out for future videos. Thank you.