 Hi, this is a video about rational expressions. First off, rational expression is literally a fraction where the top and the bottom are polynomials or just numbers. One important thing to note is that if you have a minus sign applied to the numerator or a minus sign applied to the entire denominator, that minus sign could very well just be brought out front of the fraction. Next, the domain is all x values or variable values, which the expression is defined, since rational expressions are fractions, we have to exclude any values which cause the denominator to equal zero. So we find the domain of each of the following. For part a, since x minus three is what's in the denominator, we have to exclude wherever x minus three is equal to zero. So we have to exclude x equals three, three minus three is zero. So your domain would have to be all real numbers x such that x cannot equal three. Part b, we have to exclude wherever five x is equal to zero. Once you're timesing by five, you divide both sides by five and you get zero. So your domain would have to be and set builder notation all real numbers x such that x cannot equal zero. In part c, we'll have a little bit of work to do here because as you can see, when we set x squared minus three x plus two equal to zero, it's a quadratic equation. So this one, we can factor it, so that's what we'll do. So in this situation, how to solve a quadratic equation, set equal to zero, that's factored is we have to set x minus two equal to zero or x minus one equal to zero, only then what are the product of the two b zero, if one or both of those are zero. So it looks like the two numbers that have to be excluded from the domain would have to be two and one. So the domain would be all real numbers x such that x cannot equal one or two. So that's how to find the domain. How to simplify rational expressions is that you want to make sure you factor, factor, factor, factor as much as possible, then you will cancel out common factors. Let's start off with an example two part a with 15 over 20, a basic regular fraction. Both the top and bottom are both divisible by five. So dividing the top and bottom both by five will use me with three-fourths. That's one way to get the answer or another way to get the answer is to use the factoring approach that we're about to really use a whole bunch. So 15 over 20, 15 can be broken up into three times five and 20 can be broken up into four times five. Each have a common factor of five that can be canceled out. This leaves you a three-fourth as your answer. Same answer, different strategy. In part B, you can rewrite addition in any order you want. So two x plus one can be written as one plus two x. This is also what's in the bottom. So anything divided by itself will always give you one. For C, we will do some factoring. Five x minus five has a GCF of five leading you with x minus one. X cubed minus x squared on the bottom, you can factor out x squared leading you with x minus one. X minus one on the top cancels out with x minus one on the bottom. That's five over x squared as your answer. Let's do a few more. In part E, we'll factor the numerator to get x plus two times x plus two. They'll definitely be at an advantage if you have practiced your factoring skills and are pretty good with them. On the bottom, I can pull out the greatest common factor of x. One factor of x plus two on top cancels out one factor of x plus two on the bottom. You're allowed to have x plus two over x. Part E, the top and bottom look the same except on the top, x is positive. On the bottom, x is negative. On the top, seven is negative. On the bottom, seven is positive. The signs are flipped. One strategy I'm going to do is I'm going to factor out a negative sign out of the bottom which causes me to have negative seven plus x in parentheses. You can rewrite the negative seven plus x. You can reorder them into x minus seven. Only then will you then see that x minus seven on top cancels with x minus seven on the bottom, leaving you with just a negative. A negative one is what that is. That's simplifying rational expressions. We'll be doing a lot of that same process as we jump into multiplying rational expressions. We'll still completely factor the fractions that we see. We'll multiply the numerators and denominators together and then we'll simplify. So in part A, recall how to multiply fractions together. You multiply the tops together. You multiply the bottoms together. So on top, I have three times ten. On the bottom, I have five times 11, 30 over 55. Divide top and bottom both by five to give you six over 11. One other method that you might have learned would be instead of waiting till the end to simplify, which is after you multiply the tops and bottoms together, you could actually cross simplify, cancel out diagonally. So ten and five are both divisible by five. So divide ten by five to get two. Divide five by five to get one. You have three times two on top. One times 11 on the bottom. And look, we still get six over 11. All right, part B. I'm going to write x plus seven squared as x plus seven times x plus seven, so we can see it a little better. The denominator cannot be factored in the first fraction. It's left as x minus seven. All right, in the second fraction, x is on top. And on the bottom, x squared plus seven x, we can factor out x, leaving us with x plus seven. Diagonally, I can cancel out a factor of x plus seven from the top of the first fraction with the factor of x plus seven on the bottom of the second fraction. Furthermore, in the second fraction, I can cancel out x on top with the x on the bottom, leaving me with x plus seven over x minus seven. So take note that we can cancel out vertically or we can cancel out diagonally. Both are completely fine. So we'll continue this more a little bit later, but for now, thanks for watching.