Example 1

Step 1: Express each term using prime factorization.

\(6x^2y^4z \)	\(12xy^3z \)	\(15xy^2z \)
\(=3 \cdot 2 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot z \)	\(=3 \cdot 2 \cdot 2 \cdot x \cdot y \cdot y \cdot y \cdot z \)	\(=3 \cdot 5 \cdot x \cdot y \cdot y \cdot z \)

Step 2: Identify all common factors.



Step 3: Determine the product of the common factors. This product will be the GCF.

GCF = \(3xy^2z \)

Step 4: Check the GCF by asking:

• Is \(3\) the largest factor common to \(6\), \(12\), and \(15\)?
• Are the exponents on each variable of the GCF the largest permitted given the three terms? Note: the exponents on each variable in the GCF correspond to the lowest exponents on the corresponding variables in the given terms.
  
  

  \[\frac{{6x^2 y^4 z}}{{3xy^2 z}} = 2xy^2 \]

  \[\frac{{12xy^3 z}}{{3xy^2 z}} = 4y\]

  \[\frac{{15xy^2 z}}{{3xy^2 z}} = 5\]

  
  Note that there are no factors common to \(2xy^2 \), \(4y \), and \(5 \), and thus \(3xy^2z \) is indeed the GCF.