Factoring polynomials by taking common factors
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- [Voiceover] The rectangle below has an area of 12x to the fourth plus 6x to the third plus 15x squared square meters. And we can see the area right over here and they broke it up; this green area is 12x to the fourth, this purple area is 6x to the third, this blue area is 15x squared. You add 'em all together, you get this entire rectangle which would be the combined area is 12x to the fourth plus 6x to the third plus 15x 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 12x to the fourth, 6x to the third, and 15x squared. What is the length and width of the rectangle? I encourage you to pause the video and 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 12x to the fourth, 6x third, and 15x squared, well then we can factor that out and then what we have left over is going to be the width. So let's figure out what is the greatest common monomial factor of these three terms. And the first thing we could look at is let's look at the coefficients. Let's figure out what's the greatest common factor of 12, six, and 15. And there is a couple of ways you could do it. You could do it by looking at a prime factorization, you could say alright, well 12 is two times six which is two times three, that's the prime factorization of 12. Prime factorization of six is just two times three. Prime factorization of 15 is three times five. And so the greatest common factor, the largest factor that's divisible into all of them, so let's see, we can throw a three in there. Three is divisible into all of them. And that's it 'cause we can't say a three and a two. A three and a two would be divisible into 12 and six, but there's no two that's divisible into 15. We can't say a three and a five 'cause five isn't divisible into 12 or six; so the greatest common factor is going to be three. Another way we could have done this is we could have said where are the non-prime factors of each of these numbers. 12 you could have said, OK I can get 12 by saying one times 12 or two times six or three times four. Six you could have said, let's see, that could be one times six or two times three. So those are the factors of six. And then 15 you could have said well one times 15 or three times five. And so you say the greatest common factor? Well three is the largest number that I've listed here that is common to all three of these factors. So once again, the greatest common factor of 12, six, and 15 is three. So when we're looking at the greatest common monomial factor the coefficient is going to be three. And then we look at these powers of X. We have X to the fourth. I'm using a different color. We have X to the fourth, X to the third, and X squared. Well what's the largest power of X that's divisible into all of those? Well it's going to be X squared. X squared is divisible into X to the fourth and X to the third 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 if this is 3x squared, we can then figure out what the width is. If we were to divide 12x to the fourth by 3x squared, what do we get? Well 12 divided by three is four 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, six divided by three is two. 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 three is going to be five. X squared divided by X squared is just one, so it's just gonna be five. So the width is going to be 4x squared, plus 2x, plus five. So once again, the length, we figure that out, is the greatest common monomial factor of these terms. It's 3x squared. And the width is 4x squared plus 2x, plus five. And one way to think about it is we just factored this expression over here. We could write that. Actually I wanna see the original thing. We could write that 3x squared times 4x squared, plus 2x, plus five 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 plus 15x squared.