 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Missildine. We've now arrived upon the last lecture, number 36 in our Chapter 5 unit about algebraic functions, that is rational functions, radical functions, and these other algebraically generated functions more complicated than just polynomials. This will be the seventh lecture in that chapter, which admittedly, if I'm going to keep up the Star Wars pun, I probably need at least two more lectures and admittedly, I have no idea how many Star Wars movie Disney is going to make here, so I think we'll leave it at seven right here. And so what I want to do in this very last section is introduce a technique called partial fraction decompositions that have to do with rational functions in the following way. We're probably very familiar with the algebraic problem that if we had two rational expressions to these two algebraic fractions, take like for example, 3 over x plus 4 plus 2 over x minus 3, how do you add these things together and simplify? Well, the first thing to do is we have to find a common denominator, right? So you investigate x plus 4 and x minus 3, they don't have common denominators. So in order to add them together, we have to find that common denominator, which the least common denominator here would simply just be x plus 4 times x minus 3. We've talked about that before. So we're going to reproportion things. We're going to take the 3 over x plus 4 over the times top and bottom by x minus 3 so that the denominator now agrees, okay? And we're going to do the same thing with the 2 over x minus 3. We need to times the top and bottom by x plus 4. And so now we have this common denominator in play. We can add the numerators together. We'll distribute it and we get 3x and a negative 9, like so, distribute the 2 here, we get 2x and an 8. Because we have a common denominator, we can now add these together. So you have the 3x minus 9 plus 2x plus 8 combining like terms, 3x plus 2x is a 5x and negative 9 plus 8 is a negative 1. We then get the rational function, the numerator then. And in the denominator, if you want to, I actually abhor multiplying out denominators of a rational function, but oftentimes we're trying to do that for the worst reasons ever. x plus 4 times x minus 3, if you were to foil that out, you get x squared minus 3x plus 4x minus 12, which combines to be x squared plus x minus 12. That's great. And so this is how we can add together fractions, these rational functions. So we have these two fractions right here and we add them together to get this one right here. What if we wanted to reverse the process? What if we don't know any of this? We can't see it now, right? What if we started off with the rational function 5x minus 1 over x squared plus x minus 12? What if we took this fraction, this rational function? What if we wanted to decompose it into simpler fractions? That is fractions with simpler denominators. The term we use are partial fractions because it's fraction, which is a part of the original one. How could we break this rational function up into these partial fractions right here? How could we go from this more complicated fraction to these two right here? It's not as simple as it seems, but it's a very important skill to do algebraically. It's a purely algebraic skill, but it's very helpful. For example, in calculus when you learn about integrals and such, and so I wanna talk about in this lecture how we can compute the partial fraction decomposition. How do you find these simpler fractions? And the way we're gonna break this up is we're gonna break it up between so-called improper fractions and proper fractions. You're used to this with like integer fractions, right? If you take three-fifths, this is what we call a proper fraction because the numerator is smaller than the denominator, right? On the other hand, if you take something like seven-fifths, this is an improper fraction because you could rewrite it as a mixed number so we could write this as one and two-fifths. Or if you take like five-fifths itself or like 10-fifths, these are also improper fractions because we could simplify them, right? So whenever the numerator on top is greater than or equal to the denominator, we say it's improper. If the numerator is smaller than the denominator, we call it proper. The same thing can be done for rational functions as well. So we say that the rational expression px over qx is improper if the numerator has a degree greater than or equal to the denominator's degree. So when we say that the top is bigger than the bottom, we're talking about the degree of the polynomial. If the top has a smaller degree than the denominator, then we say it's a proper fraction. So the first step when it comes to a partial fraction decomposition is to determine whether the fraction and play is proper or not. If it's a proper fraction, we can move on to the next step of the decomposition process. If it is an improper fraction, then we have to do polynomial division. So take as an example here, r of x here, it will be 4x over x squared minus three. Notice the numerator is degree x, that is it's degree one, it's a linear polynomial, the bottom is a degree two polynomial. Previously in this series, this is what we would call a bottom heavy fraction. Bottom heavy fractions are what we mean by proper fractions, that's what we want here. So this one is proper, so we would proceed to do the next step, which we haven't talked about yet. Okay, we don't have to worry about anything there. Now if you look at the next example, this one actually is an improper fraction, right? You'll notice that the numerator has degree three, the denominator has degree one, three is bigger than one, so this is an improper fraction. So what we have to then do is we wanna convert our improper fraction into a mixed number, so to speak. And how does this process work? Well, five goes into seven, right? Five goes into seven once, we subtract the five, we get a remainder of two, right? And so here's the quotient, here's the remainder. We do the same thing for polynomials, that's what long division's all about here. So we're gonna take the numerator, x cubed plus three, and we're gonna divide it by the divisor, the denominator x minus one. Now admittedly this one, since the denominator's x minus one, we could use synthetic division, but I'm just gonna do long division, because that's generally what you have to do here. So x goes into x cubed x squared times, if we take x squared and times it by x minus one, you're gonna end up with x cubed minus x squared. If you distribute a negative one through that, you're gonna get negative x cubed plus x squared. So x cubed minus x squared, those cancel out, we're gonna bring down the x, so we get x squared plus x, and so we repeat the process, x divides into x squared, x many times, because I'm just taking x squared divided by x here to get x, that's what that is. So then, we then are gonna take x times x minus one, that gives us x squared minus x times that by negative one, we then subtract that from below right here, x squared plus x, we see that the x squared's canceled, the x is actually double up, so we get a two x, there's nothing to bring down, so it's just a zero. So then we repeat the process one more time, how many times does x divide into two x? That'll give us two times, right? Two times x minus one is gonna be two x minus two, which when we subtract that, we're gonna get two x plus two, the two x is canceled, we have a remainder of two. So with this remainder of two right here, we can then write the rational function, the improper rational function, x cubed plus x over x minus one, we can write that as the quotient x squared plus x plus two, and then we have a proper rational function part, this is now proper, this rational function could sit with the queen, in which case we have two over x minus one. And so the first step when it comes to a partial fraction decomposition is make sure the rational function is proper. If it's an improper fraction, do long division, you'll get a polynomial plus a proper rational part.