Final answer:
The greatest common factor of the monomials 57x³y⁵, 51x³y⁶, and 3x²y² is 3x²y².
Step-by-step explanation:
The goal is to find the greatest common factor (GCF) of the three monomials 57x³y⁵, 51x³y⁶, and 3x²y². To find the GCF of monomials, we need to look at both the coefficients and the variables separately.
Firstly, let's find the GCF of the coefficients 57, 51, and 3. The greatest divisor common to all three numbers is 3 since it's the only divisor that can divide evenly into all of them.
For variables, we look at the lowest power of x and y in all the monomials. Since all monomials have 'x' raised at least to the power of 2 and 'y' raised at least to the power of 2, the GCF will include x² and y².
Thus, the greatest common factor of the given monomials is 3x²y².