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# Factoring Monomials

Did you know that it's possible to break down a monomial into its prime factors the same way you can break down any whole number into its prime factors? It is, and in this article, we'll look at how.

## The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be expressed in the form of primes. Basically, all of the natural numbers can be expressed in the form of the product of their prime factors. Remember that a prime number is a number that is divisible by 1 and itself only.

An example of a prime number is 13. It can only be divided by 1 and itself.

The number 26, on the other hand, is not a prime number. It can be factorized into its primes:

$13*2=26$

## The prime factorization of monomials

The prime factorization of a monomial is an expression of its prime numbers, single variables, and possibly a -1.

Example 1

Find the prime factorization of the monomial $-45{x}^{3}y{z}^{5}$ .

First, find the prime factors for 45. Start with $5*9$ . Then 9 can be broken down further into $3*3$ , making $3*3*5$ the prime factorization of 45.

Then write the powers out individually and add a -1 to the beginning to make the whole thing negative.

$-1*3*3*5*x*x*x*y*z*z*z*z*z$

Example 2

Find the prime factorization for the monomial $32p{q}^{3}{r}^{2}$ .

First, find the prime factors for 32. Start with $2*16$ . That breaks down to $2*2*8$ . That breaks down to $2*2*2*4$ . That breaks down to $2*2*2*2*2$ , and that's the prime factorization for 32.

Then write out the powers individually, and this time, there is no need to add a -1 because the monomial is not a negative monomial.

$2*2*2*2*2*p*q*q*q*r*r$

## Flashcards covering the Factoring Monomials

Algebra II Flashcards