f of x is equal to 2x

squared plus 15x minus 8. g of x is equal to x

squared plus 10x plus 16. Find f/g of x. Or you could interpret this

is as f divided by g of x. And so based on the

way I just said it, you have a sense

of what this means. f/g, or f divided by g, of x,

by definition, this is just another way to write f

of x divided by g of x. You could view this as

a function, a function of x that’s defined

by dividing f of x by g of x, by creating

a rational expression where f of x is in the numerator and

g of x is in the denominator. And so this is going to be equal

to f of x– we have right up here– is 2x

squared 15x minus 8. And g of x– I will do in blue–

is right over here, g of x. So this is all

going to be over g of x, which is x squared

plus 10x plus 16. And you could leave it this

way, or you could actually try to simplify

this a little bit. And the easiest way

to simplify this would see if we could factor the

numerator and the denominator expressions into maybe

simpler expressions. And maybe some of them might be

on– maybe both the numerator and denominator is divisible

by the same expression. So let’s try to

factor each of them. So first, let’s

try the numerator. And I’ll actually do it up here. So let’s do it. Actually, I’ll do it down here. So if I’m looking at 2x

squared plus 15x minus 8, we have a quadratic expression

where the coefficient is not 1. And so one technique to factor

this is to factor by grouping. You could also use

the quadratic formula. And when you factor

by grouping, you’re going to split up

this term, this 15x. And you’re going to

split up into two terms where the coefficients

are, if I were to take the product

of those coefficients, they’re going to be

equal to the product of the first and the last terms. And we proved that

in other videos. So essentially, we want

to think of two numbers that add up to 15,

but whose product is equal to negative 16. And this is just the technique

of factoring by grouping. It’s really just an attempt to

simplify this right over here. So what two numbers that,

if I take their product, I get negative 16. But if I add them, I get 15? Well, if I take the product

and get a negative number, that means they have to

have a different sign. And so that means one of

them is going to be positive, one of them is going

to be negative, which means one of them

is going to be larger than 15 and one of them is

going to be smaller than 15. And the most obvious

one there might be 16, positive

16, and negative 1. If I multiply these two things,

I definitely get negative 16. If I add these two things,

I definitely get 15. So what we can do is

we can split this. We can rewrite this expression

as 2x squared plus 2x squared plus 16x minus x minus 8. All I did here is I

took this middle term and, using this technique

right over here, I split it into

16x minus x, which is clearly still just 15x. Now what’s useful

about this– and this is why we call it factoring

by grouping– is we can see, are there any common factors

in these first two terms right over here? Well, both 2x squared and 16x,

they are both divisible by 2x. So you could factor out a

2x of these first two terms. This is the same thing as

2x times x plus x plus 8. 16 divided by 2 is 8,

x divided by x is 1. So this is 2x times x plus 8. And then the second two

terms right over here– this is the whole

basis of factoring by grouping– we can

factor out a negative 1. So this is equal to

negative 1 times x plus 8. And what’s neat here is

now we have two terms. Both of them have

an x plus 8 in them. So we can factor

out an x plus 8. So if we factor out

an x plus 8, we’re left with 2x minus 1, put

parentheses around it, times the factored out x plus 8. So we’ve simplified

the numerator. The numerator can be rewritten. And you could have gotten here

using the quadratic formula as well. The numerator is 2x

minus 1 times x plus 8. And now see if you can

factor the denominator. And this one’s more

straightforward. The coefficient here is 1. So we just have to

think of two numbers that when I multiply

them, I get 16. And when I add them, I get 10. And the obvious one is 8 and

2, positive 8 and positive 2. So we can write this as

x plus 2 times x plus 8. And now, we can simplify it. We can divide the numerator

and denominator by x plus 8, assuming that x does

not equal negative 8. Because this function

right over here that’s defined by

f divided by g, it is not defined when

g of x is equal to 0, because then you have

something divided by 0. And the only times that

g of x is equal to 0 is when x is equal to negative

2 or x is equal to negative 8. So if we divide the numerator

and the denominator by x plus 8 to simplify it, in order to not

change the function definition, we have to still put the

constraint that x cannot be equal to negative 8. That the original function,

in order to not change it– because if I just cancelled

these two things out, the new function with these

canceled would be defined when x is equal to negative 8. But we want this

simplified thing to be the same exact function. And this exact

function is not defined when x is equal to negative 8. So now we can write f/g

of x, which is really just f of x divided by g of

x, is equal to 2x minus 1 over x plus 2. You have to put the condition

there that x cannot be equal to negative 8. If you lost this

condition, then it won’t be the exact

same function as this, because this is not defined

when x is equal to negative 8.

Thanks you this vid

It helped he alot

@keyloggerable1

They are definitely not random. I guess it's something like "unit 17, lesson 3, topic 1, whatever 4". It would be there to identify the videos automatically (the videos are embedded in learning platform) or at least to identify them after a long time.

2:00 It's the sum of the coefficients, you might want to put an annotation.

I have to ask: function of functions?

@keyloggerable1 I THINK they're the khanacademy website id code…

can anyone tell me what happens if x=-2?

The function will be undefined. But you can already see that with the final answer 2x-1/x+2. The reason we have to specify the -8 is because the last version of the answer 2x-1/x+2 does not tell us that -8 could also lead to the undefined function before the last simplification.

4:05 why did he just remove an 8?

and can someone tell me where her covered that splitting up the 15x

he removes the 8 because there are two instances of x + 8, thus canceling out.

So he basically found the limit of f/g when x approaches -8.

I STILL CAN'T GET IT.

Can someone clarify why he removes -8 but not -2? They would both make the function undefined, so why only remove -8?

Still can't get it

wonderful. really helpful

I understand nothing. Is this applicable in quadratic equation rather?

Thank you for this video helps a lot ❤

I see this and wonder where I’m going to use this outside of school ?

I’m gonna fail my midterm exam! I still can’t get it! Am I too slow to not understand how to solve it?