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Current time:0:00Total duration:5:27

Video transcript

the rectangle below has an area of 12 X to the fourth plus six X to the third plus 15 x squared square meters and we can see the area right over here they broke it up this green area is 12 X to the fourth this purple area is 6 X to the third this blue area is 15 x squared you add them all together you get this entire rectangle which would be the combined area is 12 X to the 4th plus 6 X to the 3rd plus 15 x squared the length of the rectangle in meters so this is the length right over here that we're talking about so we're talking about this distance the length of the rectangle in meters is equal to the greatest common monomial factor of 12 X to the 4th 6 X to the 3rd + 15 x squared what is the length and width of the rectangle I encourage you to pause the video and try to think try to work through it on your own well the key realization here is that the length times the width the length times the width is going to be equal to this area and if the length is the greatest common monomial factor of these terms of 12 X to the 4th 6 X 3rd + 15 x squared well then we can factor that out and then what we have leftover is going to be is going to be the width so let's figure out what is the greatest common monomial factor of these of these three of these three terms and the first thing we can look at is let's look at the coefficients let's figure out what's the greatest common factor of 12 6 and 15 and there's a couple of ways you could do it you could do it by looking at a prime factorization you could say all right well the 12 is 2 times 6 which is 2 times 3 that's the prime factorization of 12 prime factorization of 6 is just 2 times 3 prime factorization of 15 is 3 times 5 and so the greatest common factor the largest the largest factor that's divisible into all of them so let's see we can throw a we can throw a 3 in there 3 is divisible into all of them and that's it because we can't say a 3 and a 2 with would be divisible into 12 and six but there's no two that's divisible into 15 we can't say 3 into 5 because 5 isn't divisible into 12 or 6 so the greatest common factor is going to be 3 another way we could have done this is we could said what are the magazine non prime factors of each of these numbers 12 you could have said okay I can get 12 by saying 1 times 12 or 2 times 6 or 3 times 4 6 you could have said let's see that could be 1 times 6 or 2 times 3 so those are the factors of 6 and then 15 you could have said well 1 times 15 or 3 times 5 and so you say the greatest common factor well 3 is the is the largest number that I've listed here that is common to all three of these factors so once again the greatest common - common factor of 12 6 + 15 is 3 so we're looking at the greatest common monomial factor the coefficient is going to be 3 and then we look at these powers of X we have X to the fourth we have X to the use in a different color we have X to the fourth X to the third and x squared well what's the largest power of XS divisible into all of those it's going to be x squared x squared is divisible into X to the fourth and X desert and of course x squared itself so the greatest common monomial factor is 3x squared this length right over here this is 3x squared so this is 3x squared we can then figure out what the width is so what's if we if we were to divide 12x to the fourth by 3x squared what do we get well 12 divided by 3 is 4 and X to the fourth divided by x squared is x squared notice 3x squared times 4x squared is 12x to the fourth and then we move over to this purple section if we take 6x to the third divided by 3x squared 6 divided by 3 is 2 and then X to the third divided by x squared is just going to be X and then last but not least we have 15 divided by 3 is going to be 5 x squared divided by X square is just 1 so it's just going to be 5 so the width is going to be 4x squared plus 2x plus 5 so once again the length we figure that out it was the greatest common monomial factor of these terms 3x squared and the width is 4x squared plus 2x plus 5 and one way to think about is we just factored we just factored this expression over here we could write we could write that actually I want to see the original thing we could write that 3x squared times 4x squared plus 2x plus 5 which is the entire width well that's going to be equal to the area that's going to be equal to our original expression 12x to the fourth power plus 6x to the third + 15 x squared