To determine which expressions are factors of the polynomial represented by 500x3 * 108y18 * 6 * 5x * 3y6 * 25x2 * 15xy6 * 9y2, we can first simplify the expression by examining how a factorization works.
1. **Prime Factorization:** For each of the constants in the expression, we factor them into their prime components:
- 500 = 22 * 53
- 108 = 22 * 33
- 6 = 2 * 3
- 5 = 5
- 3 = 3
- 25 = 52
- 15 = 3 * 5
- 9 = 32
By combining these, we get:
500 * 108 * 6 * 5 * 3 * 25 * 15 * 9 = (22 * 53) * (22 * 33) * (2 * 3) * (5) * (3) * (52) * (3 * 5) * (32) = 25 * 38 * 56 = 86400*56
2. **Variable Parts:** Now we also consider the variables. We have:
- x: The highest power is x3 + x + x 2 + x = x3 + 1 + 2 + 1 = x7
- y: The highest power is y18 + y6 + y6 + y2 = y18 + 6 + 6 + 2 = y32
3. **Combining Factors:** Thus the complete factorization can be determined as:
86400 * (x * y)45, which shows that there are various factors of this expression.
4. **Conclusion:** Since any number which can be formed from these primes up to those powers and combinations (like pulling out a single term, certain combinations of x and y) can be a factor of this whole expression. Therefore, all options given in the question are indeed factors of this expression.