Question 1

The greatest common factor of the three terms is 6x²y.

The greatest common factor (GCF) of the three terms in the polynomial 12x^4y^2 + 18x²y^4 + 24x³y is 6x²y. This is determined by identifying the highest common factors of the coefficients and the variables in each term.

A) 2xy

2xy is not the greatest common factor because while it is a common factor of the terms, it does not account for the highest coefficients and powers present in all three terms. The coefficients can be factored further, and the variables can also be raised to higher powers, making 2xy insufficient.

B) 2x²y

2x²y is a common factor of the terms, but it is not the greatest. The coefficients of the terms can be factored down to a larger GCF, and thus, while 2x²y appears in each term, it does not encompass the maximum values of the coefficients or the variables.

C) 3x^4y^4

3x^4y^4 cannot be the GCF because it exceeds the powers of variables present in the other terms. Specifically, the term 18x²y^4 has a lower power of x and the term 24x³y has a lower power of y. Therefore, this option is incorrect.

D) 6x²y

6x²y is indeed the greatest common factor as it includes the highest common coefficient (6) and the lowest powers of each variable (x² and y) present in all three terms. This reflects the maximum shared factors among the polynomial's terms.

E) 6x⁹y^7

6x⁹y^7 is not a viable option as it exceeds the powers of x and y found in any of the terms. The highest powers of x and y in the given polynomial do not reach 9 and 7, respectively, making this choice incorrect.

Conclusion

The correct answer, 6x²y, effectively captures the highest common factors of both the coefficients and the variable powers across the polynomial's terms. All other options fail to match this GCF either by being too small or exceeding the limits of the variables involved. Therefore, 6x²y is the definitive greatest common factor.