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CCSS Math: HSF.BF.A.1b

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.