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Multiplying binomials by polynomials: area model

Video transcript

what I want to do in this video is figure out multiple ways to express the area of this entire large rectangle which we see is made up of these six smaller rectangles so there's a couple of ways that we can do it one way is we can just multiply the height of this big rectangle times the width of this big rectangle so what's its height well from here to here that distance is going to be Y squared and then from there to there that distance is going to be negative 6y and I know what you're thinking how can my distance be negative 6y is it's a distance always positive well even negative 6y can be positive if Y is negative so it's completely reasonable to say well this distance could be negative 6y so this entire the entire height right over here is going to be it's going to be Y squared minus 6y or you could view it as Y squared plus this distance which is negative 6y y squared plus negative 6y which is the same thing as y squared minus 6y so that's the height of the of this big rectangle what's its width well the width is going to be the width of this purple rectangle it's going to be 3y squared plus the width of this yellow rectangle which is negative 2y and that can have a negative out here or the same logic why this could have a negative or the negative 6y could have a negative and then plus the width of this blue rectangle and so if you add them all together the width of the entire rectangle is going to be 3y squared minus 2i minus 2y plus 1 and just like that this expression that I just wrote down will give us the area for the entire the area for the entire big rectangle now there's another way to do it in a big clue was that we subdivided the big rectangle into these six smaller rectangles and we have the dimensions for the six smaller rectangles and so we could find the area for each of these and then we can add them all together so let's look at this first one height times width the area of this purple rectangle is going to be the height I squared times the width which is 3y squared which is going to be equal to it's going to be 3 and then y squared times y squared is y to the fourth power what's the area of the CLO rectangle height is y squared it's going to be y squared times the width times negative 2y which is going to give us negative 2y to the third power what about the blue one well height times width it's going to be Y squared times one which of course is just going to be equal to Y squared now this green one it's going to be the height which is now negative 6y times the width which is 3y squared which is going to be equal to see negative 6 times 3 is negative 18 and then Y times y squared is y to the third power now the area of this grey rectangle it's going to be the height which is negative 6y times the width which is negative 2y which gets us negative 6 times negative 2 is positive 12 y times y is y squared and then finally the area of this rectangle right over here it's going to be the height which is negative 6y times the width which is just 1 which is equal to negative 6y and so if we want the area of this entire rectangle we can just add up the areas of the smaller ones so it's going to be equal to the 3 it's going to be equal to the 3y to the fourth 3y to the fourth plus negative 2y to the third power negative let me write this in a color that corresponds to that negative 2y to the third power plus y squared plus y squared minus 18 y to the third power so minus 18 y to the third power + 12 y squared let's try that in black so plus 12y squared and then last but at least we have the minus 6 y minus 6 y so this is an expression for the in area of the entire thing but we can simplify it more let's see we have we only have one fourth degree term so I'll just rewrite that so we have we have one fourth degree term so I'm going to rewrite that 3y to the fourth power now how many third degree terms do we have we have negative 2y to the third power we have negative 18 y to the third power so if we add these two together how many Y to the third powers do we have well negative two plus negative 18 is negative 20 negative 20 y to the third power and then how many second degree terms do we have well we have 1y squared right over here and then we have another 12 Y Squared's you add those together you're going to have 13 Y Squared's and then finally we still have this we still need to subtract the 6y and there you have it another expression for the area of the entire rectangle and the whole point of doing this is to realize that this up here and this down here are equivalent and that the way that we multiply this actually corresponds to exactly how we found the areas of the smaller rectangles right over here you would say Y squared times 3y squared is 3y fourth Y squared times negative 2y is negative 2y to the third power y squared plus one y squared times one is y squared which is exactly what we did when we found the area of these rectangles and this as you say in this top row and then you would take the negative six and you would say negative six times 3y squared is negative 18 Y to the third negative six times negative two Y is positive 12 y squared negative six times one is or negative six Y times one is negative six Y and just to realize that this isn't just some type of Voodoo that we're doing it completely makes sense when you think about it in terms of an area model like this