Simplifying rational expressions
Simplify the rational expression and state the domain. Once again, we have a trinomial over a trinomial. To see if we can simplify them, we need to factor both of them. That's also going to help us figure out the domain. The domain is essentially figuring out all of the valid x's that we can put into this expression and not get something that's undefined. Let's factor the numerator and the denominator. So let's start with the numerator there, and since we have a 2 out front, factoring by grouping will probably be the best way to go, so let's just rewrite it here. I'm just working on the numerator right now. 2x squared plus 13x plus 20. We need to find two numbers, a and b, that if I multiply them, a times b, needs to be equal to-- let me write it over here on the right. a times b needs to be equal to 2 times 20, so it has to be equal to positive 40. And then a plus b has to be equal to 13. The numbers that jump out at me immediately are 5 and 8, right? 5 times 8 is 40. 5 plus 8 is 13. We can break this 13x into a 5x and an 8x, and so we can rewrite this as 2x squared. It'll break up the 13x into-- I'm going to write the 8x first. I'm going to write 8x plus 5x. The reason why I wrote the 8x first is because the 8 shares common factors with the 2, so maybe we can factor out a 2x here. It'll simplify it a little bit. 5 shares factors with the 20, so let's see where this goes. We finally have a plus 20 here, and now we can group them. That's the whole point of factoring by grouping. You group these first two characters right here. Let's factor out a 2x, so this would become 2x times-- well, 2x squared divided by 2x is just going to be x. 8x divided by 2x is going to be plus 4. Let's group these two characters. And if we factor out a 5, what do we get? We get plus 5 times x plus 4. 5x divided by 5 is x, 20 divided by 5 is 4. We have an x plus 4 in both cases, so we can factor that out. We have x plus 4 times two terms. We can undistribute it. This thing over here will be x plus four times-- let me do it in that same color-- 2x plus 5. And we've factored this numerator expression right there. Now, let's do the same thing with the denominator expression. I'll do that in a different-- I don't want to run out of colors. So the denominator is right over here, let's do the same exercise with it. We have 2x squared plus 17x plus 30. Let's look for an a and a b. When I multiply them, I get 2 times 30, which is 60. And an a plus a b, when I add them, I get 17. Once again, 5 and 12 seem to work. So let's split this up. Let's split this up into 2x squared. We're going to split up the 17x into a 12x plus a 5x and that adds up to 17x. When you multiply 12 times 5, you get 60, and then plus 30. Then on this first group right here, we can factor out a 2x, so if you factor out a 2x, you get 2x times x plus 6. In that second group, we can factor out a 5, so you get plus 5 times x plus 6. Now, we can factor out an x plus 6, and we get we get x plus 6 times 2x plus 5. We've now factored the numerator and the denominator. Let's rewrite both of these expressions or write this entire rational expression with the numerator and the denominator factored. The numerator is going to be equal to x plus 4 times 2x plus 5. We figured that out right there. And then the denominator is x plus 6 times 2x plus 5. It might already jump out at you that you have 2x plus 5 in the numerator and the denominator, and we can cancel them out. We will cancel them out. But before we do that, let's work on the second part of this question. State the domain. What are the valid x values that we could put in here? A more interesting question is what are the x values that will make this rational expression undefined? It's the x values that will make the denominator equal to 0, and when will the denominator equal to 0? Well, either when x plus 6 is equal to 0, or when 2x plus 5 is equal to 0. We could just solve for x here. Subtract 6 from both sides, and you get x is equal to negative 6. If you subtract 5 from both sides, you get 2x is equal to negative 5. Divide both sides by 2. You get x is equal to negative 5/2. We could say the domain-- let me write this over here. The domain is all real numbers other than or except x is equal to negative 6 and x is equal to negative 5/2. The reason why we have to exclude those is those would make this denominator-- either way you're right. It's going to go make the denominator equal to 0, and it would make the entire rational expression undefined. We've stated the domain. Now let's just simplify the rational expression. We've already said that x cannot be equal to negative 5/2 or negative 6, so let's just divide the numerator and the denominator by 2x plus 5. Or just looking at the 2x plus 5, we know that 2x plus 5 won't be 0, because x won't be equal to negative 5/2, and so we can cancel those out. The simplified rational expression is just x plus 4 over x plus 6.