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Greatest common factor of monomials

Follow along as Sal finds the greatest common factor of 10cd^2 and 25c^3d^2 and discover the secret to finding the greatest common factor of monomials! Dive into prime factorization and variable parts, and learn how to break down monomials into their simplest forms. Uncover the common factors and master the art of algebraic expressions. Created by Sal Khan and Monterey Institute for Technology and Education.

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• how is are the monomials used in life?
(22 votes)
• We use monomials all the time, particularly when counting things. 5 gallons, 12 eggs, 20 dollars, 3 kids; all of these are monomials, or single term expressions where we are multiplying a constant by a variable. An example of a simple binomial equation could be for a classroom made up of 10 boys and 9 girls which we can express as something like:

c = 10b + 9g

The funny part is that we all use a fair bit of algebra on a regular basis without even knowing it because algebra is mostly just common sense calculations boiled down into pure logic.
(39 votes)
• what dos monomial mean?
(8 votes)
• it is a one-term polynomial (think of it as a constant multiplied by a variable (x) raised to some power)
(20 votes)
• Just wondering, how do you calculate the greatest common factor of, say 56?
(4 votes)
• The greatest common factor deals with two expressions, not one, so 56 doesn't have a GCF by itself because there's not another number to compare it to. If you're talking about breaking a constant expression down into its constituent factors, all you'd have to do is find all the numbers you can multiply by to get 56. Those would be 2(28), 4(14), and 8(7).
(18 votes)
• what to do if a number is negative? for example -6t + 9t.
(6 votes)
• To find the GCF you can ignore the sign. For -6t + 9t the GCF is 3.
(6 votes)
• At , Sal Khan said "But I'll put that in quotes depending on whether c is negative or positive and d is greater than or less than 0". I think he made a mistake in this part, although I think it will also be helpful to others if I talk about why this is a mistake. The true greatest common factor does not depend on whether d is less than or equal to zero, as (-a)^2=(a)^2, as Sal Khan said, but rather on whether the absolute value of d is less than 1, in which case the absolute value of the entire monomial will decrease as x increases in d^x. For example, if d=1/3, then d^3 would be less than d^4, as d^3=1/27, and d^4=1/81. Now, if |d| is greater than 1, as x's value increases, it is true that the absolute value of the monomial 5cd^x's will increase, provided c and d are both non-zero numbers and |d| is not equal to 1. However, this does not translate to "If |d| is greater than 1 then as x's value increases, the value of 5cd^x will increase". This may or may not be true under certain circumstances. If c is positive, then yes, the value of 5cd^x will increase when x's value increases. However, if c is negative, the value of 5cd^x will only decrease when x's value increases. Now, using this knowledge, to know whether if 5cd^2 is truly greater than 5c by itself, requires knowledge of if c is negative or positive, and if |d| is less than 1. So, we have 4 different cases. I will use + for positive, - for negative, > for the absolute value is greater than 1, and < for the absolute value is less than 1. Increasing means that the value of the monomial increases from 5c to 5cd^2. (c+,d<)-Decreasing. (c-,d<)-Increasing. (c+,d>)-Increasing. (c+,d<)-Decreasing. Although Khan did say the part about c correctly, that the value of the monomial depended on whether c was negative or positive, I do believe that he meant to say "and d is greater than or less than 1" instead of "and d is greater than or less than 0".
(6 votes)
• Well...What's the difference between a polynomial and a monomial?
(4 votes)
• Also, polynomials can have any number of terms, as long as it is more than one.
(2 votes)
• Is there any other way like Saxon's technique?
(5 votes)
• is there another way to do this?
(4 votes)
• When he found the solution, I realized that he could have divided the integer and subtracted the powers of c and d (or just c in this case.) Is it okay if we do just that instead of breaking it down into more steps?
(3 votes)
• You can break it down into more steps as long as they're correct.
(2 votes)
• If you can find the greatest common factor of two monomials, can't you find the least common multiple of two monomials using the same way to do it?
(3 votes)

Video transcript

Find the greatest common factor of these monomials. Now, the greatest common factor of anything is the largest factor that's divisible into both. If we're talking about just pure numbers, into both numbers, or in this case, into both monomials. Now, we have to be a little bit careful when we talk about greatest in the context of algebraic expressions like this. Because it's greatest from the point of view that it includes the most factors of each of these monomials. It's not necessarily the greatest possible number because maybe some of these variables could take on negative values, maybe they're taking on values less than 1. So if you square it, it's actually going to become a smaller number. But I think without getting too much into the weeds there, I think if we just kind of run through the process of it, you'll understand it a little bit better. So to find the greatest common factor, let's just essentially break down each of these numbers into what we could call their prime factorization. But it's kind of a combination of the prime factorization of the numeric parts of the number, plus essentially the factorization of the variable parts. If we were to write 10cd squared, we can rewrite that as the product of the prime factors of 10. The prime factorization of 10 is just 2 times 5. Those are both prime numbers. So 10 can be broken down as 2 times 5. c can only be broken down by c. We don't know anything else that c can be broken into. So 2 times 5 times c. But then the d squared can be rewritten as d times d. This is what I mean by writing this monomial essentially as the product of its constituents. For the numeric part of it, it's the constituents of the prime factors. And for the rest of it, we're just kind of expanding out the exponents. Now, let's do that for 25c to the third d squared. So 25 right here, that's 5 times 5. So this is equal to 5 times 5. And then c to the third, that's times c times c times c. And then d squared, times d squared. d squared is times d times d. So what's their greatest common factor in this context? Well, they both have at least one 5. Then they both have at least one c over here. So let's just take up one of the c's right over there. And then they both have two d's. So the greatest common factor in this context, the greatest common factor of these two monomials is going to be the factors that they have in common. So it's going to be equal to this 5 times-- we only have one c in common, times-- and we have two d's in common, times d times d. So this is equal to 5cd squared. And so 5d squared, we can kind of view it as the greatest. But I'll put that in quotes depending on whether c is negative or positive and d is greater than or less than 0. But this is the greatest common factor of these two monomials. It's divisible into both of them, and it uses the most factors possible.