To determine the greatest common factor (GCF) of the expression \( 24x^3y^4 - 15xy^2 - 36x^2y^3 \), we start by factoring each term separately.
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Identify the coefficients:
- The coefficients are 24, -15, and -36.
- The GCF of these numbers can be found:
- The prime factorization of 24 is \( 2^3 \cdot 3^1 \).
- The prime factorization of 15 is \( 3^1 \cdot 5^1 \).
- The prime factorization of 36 is \( 2^2 \cdot 3^2 \).
- The GCF of the coefficients: \( 3^1 = 3 \).
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Identify the variables:
- The variables in each term are \( x \) and \( y \).
- The powers of \( x \) in the terms are \( x^3 \), \( x^1 \), and \( x^2 \). The lowest power is \( x^1 \).
- The powers of \( y \) in the terms are \( y^4 \), \( y^2 \), and \( y^3 \). The lowest power is \( y^2 \).
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Combine the GCF of coefficients and variables:
- The GCF for the coefficients is \( 3 \).
- The GCF for the variable \( x \) is \( x^1 = x \).
- The GCF for the variable \( y \) is \( y^2 \).
Putting it all together, the GCF of the entire expression is:
\[ 3xy^2 \]
So, the greatest common factor of the expression \( 24x^3y^4 - 15xy^2 - 36x^2y^3 \) is \( \boxed{3xy^2} \).