determine the greatest common factor of 24x^3y^4-15xy^2-36x^2y^3

Answer 1

To determine the greatest common factor (GCF) of the expression \( 24x^3y^4 - 15xy^2 - 36x^2y^3 \), we will factor out the GCF from each term.

1. Identify the coefficients of each term:

   • The coefficients are 24, -15, and -36.

2. Find the GCF of the coefficients:

   • The factors of 24 are: \( 1, 2, 3, 4, 6, 8, 12, 24 \)
   • The factors of -15 are: \( 1, 3, 5, 15 \)
   • The factors of -36 are: \( 1, 2, 3, 4, 6, 9, 12, 18, 36 \)
   • The common factors are: \( 1, 3 \)
   • Thus, the GCF of the coefficients is \( 3 \).

3. Identify the variable parts in each term:

   • For \( 24x^3y^4 \), the variable part is \( x^3y^4 \).
   • For \( -15xy^2 \), the variable part is \( xy^2 \).
   • For \( -36x^2y^3 \), the variable part is \( x^2y^3 \).

4. Find the GCF of the variable parts:

   • For the \( x \) terms:
     • The minimum exponent of \( x \) is \( 1 \) (from \( -15xy^2 \)).
   • For the \( y \) terms:
     • The minimum exponent of \( y \) is \( 2 \) (from \( -15xy^2 \)).
   • Thus, the GCF of the variable parts is \( xy^2 \).

5. Combine the GCF of the coefficients and the variable parts:

   • The GCF of the entire expression is \( 3xy^2 \).

Therefore, the greatest common factor of \( 24x^3y^4 - 15xy^2 - 36x^2y^3 \) is \( \boxed{3xy^2} \).