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CCSS.Math:

f of X is equal to 7x minus 5 G of X is equal to X to the third power plus 4x and then they asked us to find F times G of X so the first thing to realize that this notation F times G of X it's just referring to a function that is a product of f of X and G of X so by definition this notation just means f of X f of X x times G of X and then we just have to substitute f of X with this definition G of X with this definition and then multiply out these algebraic expressions so f of X is right over there and G of X G of X is right over there so let's do it so this is going to be equal to switch back to the orange color it's going to be equal to f of X which is 7x minus 5 times G of X and G of X is X to the third power plus 4 X X to the third power plus 4 X and you could we're multiplying to two expressions that each have two terms you could use foil if you like I don't like using foil because you might forget what it's even about this is really oil is really just multiple using the distributive property twice so for example you take this expression whatever you have out here if you had a nine out here or an A or an X or anything now you have 7x minus 5 if you're multiplying it times this expression you would multiply this times each term over here so when you multiply 7x minus 5 times X to the third you get and I'll write it this way you get X to the third times actually let me write it the other way you get 7x minus 5 times X to the third times X to the third and then you have plus 7x minus 5 plus 7x minus 5 times 4x times 4x and now we can do this derivative property again we're not normally used to seeing the things we distribute on the right hand side was the same exact idea we could put the X to the third here as well and when we distribute you multiply x to the third times 7x at times negative 5 X to the 3rd times 7 X is 7 X 7 X to the 4th power X to the 3rd times negative 5 is minus 5 X to the 3rd and then you do it over here you distribute the 4 X tie over the 7 X or 4x times 7x is plus 28 x squared 4x times negative 5 is minus minus 20x and let's see if we can simplify this we only have one fourth degree term one third-degree term one second-degree term and one first degree term so actually you can't simplify this anymore and we're done this is the product of those two function definitions this is f times G of X it is a new function created by multiplying the other two functions