Final answer:
The greatest common factor of 30x^3y - 45x^2y^2 and 135xy^3 is 15xy. There is an error in the options provided, as 15x^2y cannot be the GCF because the second term doesn't contain x^2. The correct choice should be 5xy, indicating a discrepancy in the choices.
Step-by-step explanation:
The greatest common factor (GCF) of two or more polynomials is the largest polynomial that divides each of the polynomials without leaving a remainder. To find the GCF of 30x^3y - 45x^2y^2 and 135xy^3, we first list the factors of the numerical coefficients (30, 45, and 135) and find the highest power of each variable that appears in every term.
- The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
- The factors of 45 are 1, 3, 5, 9, 15, 45.
- The factors of 135 are 1, 3, 5, 9, 15, 27, 45, 135.
The greatest common numerical factor is 15. Observing the variables, the smallest power of x in all terms is x, and the smallest power of y is y. Therefore, the GCF is 15xy, which corresponds to option b) 15x^2y. Note that this is a slight error as the correct mathematical GCF is indeed 15xy, not 15x2y. However, it seems option a) 5xy would be the appropriate correct answer, as 5 is a factor of all three coefficients, and xy is the highest power of the variables present in every term.