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### Course: Integrated math 3 > Unit 2

Lesson 2: Greatest common factor# 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.

## Want to join the conversation?

- how is are the monomials used in life?(23 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.(41 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)(21 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).(19 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.(7 votes)

- At2:57, 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".(7 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.(4 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)
- Technically, yes. But it would be the opposite of what you do to find the GCF. You would have to find a term that is the smallest possible, and shows that both are factors of the term.(1 vote)

## 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.