Operations with rational expressions — Harder example

- [Instructor] We're asked, "Which of the following is equivalent to 6x squared plus 5x over 3x plus one?" Pause this video and see if you can work through this before we do it together. All right, now let's work through this together. There's two ways that you could approach this. One is to do algebraic long division. So another way of rewriting this is saying, we are going to divide 3x plus one into 6x squared plus 5x, and if algebraic long division is unfamiliar to you, I encourage you to look it up on the non-SAT part of Khan Academy. But the way we look at it, we look at the highest degree term, which here is the first degree term. And we see, all right, how many times does it go into 6x squared? Well, 3x goes into 6x squared, 2x times. And so we write that in the first degree space here. So 2x and then 2x times 3x is 6x squared, and then 2x times one is equal to 2x. And then we subtract these from above. So it's very much like the long division that you learned in elementary school. It's just, we're doing with algebraic expressions now. And so these cancel out. 5x minus 2x is equal to 3x. Now, how many times does 3x plus one go into 3x? Well, one way to think about it, 3x goes into 3x one time. So let's just write that in the zeroth degree column. You could view that as a constant column. one times 3x plus one is 3x and one. Now we wanna subtract this and we are going to be left with, those go away, and you're just left with a negative one. And you can't divide 3x anymore into negative one. So you could view that as the remainder. And so this whole thing is going to be equal to 2x plus one, and then minus one over 3x plus one. Minus one over 3x plus one. And you can see that this is choice D, right over here. Now, the other way that you could do it if you forgot how to do algebraic long division, or you think this is just taking too long. Given that you have multiple choices here, you could just try out a simple number that's easy to compute. I wouldn't try out zero or one, because whether you just multiply by zero one or whether you square them, zero squared is zero, one squared is one. So it might not differentiate itself so much. So what I would do is maybe use a number like two and then see which of these other expressions are the same when you evaluate two. So for example, if I say six times two squared plus five times two, over three times two plus one, this is equal to, see, six times four, this is 24. Five times two is 10. This is six. So we get 34/7. 34/7. Now, if you put two in 2x, you just get four. That's not 34/7. Rule that one out. You put two over here, three times two is six, plus four is 10. That's not 34/7. Rule that out. You put two here, you get four minus one over, let's see, it looks like one over, three times two is six plus one is seven. So minus 1/7, and let's see, four is the same thing as 28/7 minus 1/7, which is equal to 27/7, still not 34/7. So if you're doing this on a standardized test and you feel confident in your math so far you might be able to do sets D, but we can verify that. Two times two, and once again, I've picked the number two just for simple computation, plus one minus one over three times two, plus one. So this is going to give us five right over here. And this is 1/7. So five minus 1/7. So that's the same thing as 35/7 minus 1/7, this is five, that is equal to 34/7. So when you use the number two you also see that this one evaluates to the same value. That's possible you try out some number, a two or a three, and several of them come to the same value, in which case you could rule out some, but that's where the algebraic long division is more useful. You know for sure what the choice is going to be.