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## Algebra 2

### Course: Algebra 2>Unit 3

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? •  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.
• what dos monomial mean? • Just wondering, how do you calculate the greatest common factor of, say 56? • 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).
• what to do if a number is negative? for example -6t + 9t. • Well...What's the difference between a polynomial and a monomial? • Is there any other way like Saxon's technique? • 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". • is there another way to do this?   