Final answer:
The greatest common factor for the expression 8r^5s^4 - 32r^2s is 8r^2s, determined by finding the smallest powers of the numerical coefficients and variables that are present in both terms.
Step-by-step explanation:
The greatest common factor (GCF) for the expression 8r^5s^4 - 32r^2s is the largest polynomial that evenly divides both terms of the expression.
To find the GCF, we need to factor each term to its prime factors and common variables. For the first term, 8r^5s^4, we have 2^3 for the numerical portion and r^5s^4 for the variable portion.
The second term, -32r^2s, factors into 2^5 for the numerical portion and r^2s for the variable portion.
Considering both terms, the numerical GCF is 2^3, which is the smaller power of 2 that appears in both terms.
For the variables, the GCF is r^2s since this is the smaller power of r and s that appear in both terms.
Thus, the GCF for the entire expression is 8r^2s.