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.

thyanks

Thank you for the help.(P.S. do you read the youtube comments?)

Dear Sal!

It doesnt matter which order you use when grouping. So you can write the expression as 2x^2 + 5x + 8x + 20 instead of 2x^2 + 8x + 5x + 20. After factoring the first and the second term, and factoring the third and the fourth term, you'll get the same result. And its true in every case.

High respect anyway! Great videos!

Can you not simplify the 4/6 in the end to 2/3? Or would that change the something?

gud