Determine the greatest common factor of 24x3y4−15xy2−36x2y3 :(1 point) Responses 4xy2 4 x y squared 3x2y2 3 x squared y squared 3xy2 3 x y squared xy

Answer 1

To find the greatest common factor (GCF) of the expression \( 24x^3y^4 - 15xy^2 - 36x^2y^3 \), we can first factor each term separately.

1. Coefficients: Identify the coefficients of each term:

   • The coefficients are \( 24 \), \( -15 \), and \( -36 \).

   • The GCF of the coefficients:

     • Factors of \( 24 \): \( 1, 2, 3, 4, 6, 8, 12, 24 \)
     • Factors of \( 15 \): \( 1, 3, 5, 15 \)
     • Factors of \( 36 \): \( 1, 2, 3, 4, 6, 9, 12, 18, 36 \)

   The common factors are \( 1 \) and \( 3 \), hence the GCF of the coefficients is \( 3 \).

2. Variables: Now, let's find the GCF of the variable parts.

   • For \( x \):

     • The exponents in each term are \( 3 \) (in \( 24x^3y^4 \)), \( 1 \) (in \( -15xy^2 \)), and \( 2 \) (in \( -36x^2y^3 \)).
     • The GCF for \( x \) is \( x^{\min(3, 1, 2)} = x^1 = x \).

   • For \( y \):

     • The exponents in each term are \( 4 \) (in \( 24x^3y^4 \)), \( 2 \) (in \( -15xy^2 \)), and \( 3 \) (in \( -36x^2y^3 \)).
     • The GCF for \( y \) is \( y^{\min(4, 2, 3)} = y^2 \).

3. Combining: Now, combine the GCF of the coefficients with the GCF of the variables: \[ \text{GCF} = 3xy^2 \]

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