Dividing functions

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.