 Let us write the partial fraction decomposition for the rational function 6x squared minus 22x plus 18 over x minus 1 times x minus 2 times x minus 3. Now fortunately, this is already a proper fraction, the numerator is degree 2, the denominator is degree 3, and the denominator is already factored for us, that avoids some otherwise long algebraic steps. So we're ready to look at the template of this partial fraction decomposition. Now, unlike our previous examples, the denominator here actually has three linear factors. How does that affect our template here? Well, imagine you have roommates trying to order a pizza together, right? One person's like, I want pepperoni. One person's like, I want sausage. One person's like, I want olives. We have to put them all together and make a cowboy pizza. So how that affects the template is, well, one of the partial fractions, one of the roommates, one at pepperoni, one of them wanted sausage, and one of them wanted olives. So we get a partial fraction for each of the factors in the denominator. Now we have to clear the denominators. We're going to multiply both sides of the equation by the LCD, right? Which the LCD is just the denominator of the original fraction here. Because if you take x minus 1, x minus 2, and x minus 3, the least common denominator is that cowboy pizza. It's the product of all these three things. On the left-hand side, the least common denominator, which is x minus 1 times x minus 2 times x minus 3, they will cancel leaving just the numerator, 6x squared minus 22x plus 18. When you distribute the LCD onto these pieces right here, right, you're going to see something that will look like the following. A times x minus 1, x minus 2, x minus 3. This sits above x minus 1. The x minus 1 is going to cancel, and you're going to see what's left are those factors which the partial fraction was missing from the LCD. It had an x minus 1, so that cancels out. So what's left is the stuff that it was missing. So you end up with A times x minus 2 times x minus 3, the thing it was missing. And so you're going to do that for the other pieces. We're going to get B times x minus 1 and x minus 3, that's what it was missing. And then I'm going to scooch this over a little bit. You're going to get C times x minus 1 and x minus 2. So now let's do our annihilating values here. So I'm going to start off with x equals 1. If you plug in x equals 1, B would annihilate, C would annihilate. And so on the left-hand side, you're going to get 6 minus 22 plus 18. This is equal to, you're going to get A times negative 1, 2 minus 1 there. And then you're going to get negative 2, 1 minus 3. Like I said, B and C got annihilated. And so then on the left-hand side, we have to take 6 minus 22 minus 18, right? That should just be, I mean, 6 and 18 together is a 24 minus 22 is a 2. So we get 2 is equal to positive 2A. So we see that A should equal 1. That's the first value. The next value to use to annihilate, I'm going to use x equals 2. A will be annihilated and C will be annihilated. The left-hand side will look like 6 times 2 squared, which is a 4, minus 22 times 2 plus 18. The right-hand side, we're going to get 2 minus 1, which is a 1. Then we're going to get 2 minus 3, which is a negative 1, and then we get a B. Simplifying the left-hand side, this is the hardest part here. But honestly, it's not the worst thing in the world, right? You get 6 times 4, which is 24. You get 22 times 2, which is 44 plus 18, right? The right-hand side should be a negative B. You're going to get 24 minus 44. It's a negative 20 plus 18. That gives me a negative 2, which equals negative B. And so that then tells us that B equals 2, like so. And then doing the last value, we want to annihilate using 3, right? When you do 3, that'll take away A, that'll take away B, that leaves just a C. So you have to do 6 times 3 squared minus 22 times 3 plus 18. On the right-hand side, you're going to end up with 3 minus 1, which is a 2. You get that from this factor. Then you're going to get 3 minus 2, which is 1. So you're going to get a 2C on the right-hand side. You can do this arithmetic, but I also want to remind you, you can use synthetic division if you want to, right? Remember, synthetic division, if you take 6 negative 22 and 18 and you do 3 right here, this could be a little bit easier to do. Drop down to 6. 6 times 3 is 18 minus 22 is a negative 4 times 3 is a negative 12 plus 18 would be a positive 6. That's the evaluation here. And so we're going to get that 6 equals 2C. Remember the remainder when synthetic division is done is the same thing as evaluating the polynomial, right? So you get 6 is equal to 2C or C equals 3, like so. For which then we come back up to our template, right there. And so what we saw was that our rational function was equal to 1 over x minus 1 plus 2 over x minus 2 plus 3 over x minus 3, which is our partial fraction decomposition. So if we have three linear factors in this denominator, no big deal. 4, 5, 6, we just get more and more partial fractions for all of the different factors we have in the denominator. So adjust the template and the process is otherwise the same.