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CCSS.Math:

f of X is equal to 2x squared plus 15 X minus 8 G of X is equal to x squared plus 10 X plus 16 find F over G of X or you could interpret this as f divided by G of X and so based on the way I just said it you kind of have a sense of what this means F over 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 the 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 - x squared + 15 X - 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 x squared plus 10 X 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 can factor the numerator and the denominator expressions into maybe simpler expressions and maybe some of them might be on maybe both the numerator 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 2 x squared + 15 X - 8 we have a quadratic expression where the coefficient 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 15 X and you're going to split it up into two you're going to split it up into two terms where the coefficients are if I were to take the if I were to take the product of those coefficients they're going to be equal to the product of the first of the last terms and we prove 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 a technique of factoring by grouping it's really just in 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 their 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 the big ativy one's going to be larger than fifteen and one of them is going to be smaller than fifteen and the most obvious one there might be 16 positive 16 and negative one if I multiply these two things I definitely get negative 16 if I add these two things I definitely get I definitely get 15 so we could what we can do is we can split this we can rewrite this expression as 2x squared plus 2x squared plus 16 16x minus X minus X minus 8 all I did here is I took this middle term I took this middle term and using this technique right over here I split it into 16x minus X which is clearly still just 15 X 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 + 16 X they are both divisible by 2 X so you can factor out a 2x of these first two terms so this is the same thing as 2x times X plus X plus 8 all right 16 divided by 2 is 8 X divided by X is 1 so this is 2x times X plus 8 and then these second two terms right over here and 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 if we factor out an X plus 8 we get we're left with 2x we're left with 2x minus 1 2x minus 1 I'll put parentheses around at 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 let's see if we 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 2 and positive 2 so we can write this as X plus 2 times X plus X plus 8 and now we can simplify it we can divide the numerator and the denominator by X plus 8 assuming 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 zero because then you have something divided by zero when what and the only times that G of X is equal to zero is when X is equal to negative 2 or X is equal to negative 8 so if we divide the numerator in 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 cancel these two things out the new function with these cancel would be defined when X is equal to negative 8 but we want to simplify 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 over 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 if 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