To determine which of the given choices are factors of the polynomial [tex]\( 500 x^3 + 108 y^{18} \)[/tex], we will check each choice one by one. Here's a breakdown of each factor:
### Checking if 6 is a factor:
For [tex]\( 6 \)[/tex] to be a factor of [tex]\( 500 x^3 + 108 y^{18} \)[/tex], each term in the polynomial must be divisible by 6.
- [tex]\( 500 x^3 \)[/tex] needs to be divisible by 6:
[tex]\( 500 \)[/tex] divided by 6 gives a remainder of 2, so [tex]\( 500 x^3 \)[/tex] is not divisible by 6.
- [tex]\( 108 y^{18} \)[/tex] needs to be divisible by 6:
[tex]\( 108 \)[/tex] divided by 6 equals 18, which is an integer, so [tex]\( 108 y^{18} \)[/tex] is divisible by 6.
Since not all terms in [tex]\( 500 x^3 + 108 y^{18} \)[/tex] are divisible by 6, we conclude that 6 is not a factor.
### Checking if [tex]\( 5x + 3y^6 \)[/tex] is a factor:
For [tex]\( 5x + 3y^6 \)[/tex] to be a factor of [tex]\( 500 x^3 + 108 y^{18} \)[/tex], the polynomial must be exactly divisible by [tex]\( 5x + 3y^6 \)[/tex].
- [tex]\( 500 x^3 \)[/tex] divided by [tex]\( 5x \)[/tex] is [tex]\( 100 x^2 \)[/tex], every corresponding term in the polynomial must cancel out perfectly with multiplication.
- [tex]\( 108 y^{18} \)[/tex] divided by [tex]\( 3 y^6 \)[/tex] is [tex]\( 36 y^{12} \)[/tex], every corresponding term in the polynomial must also cancel out perfectly with multiplication.
Since [tex]\( 500 x^3 \)[/tex] and [tex]\( 108 y^{18} \)[/tex] do not yield a polynomial that when multiplied with [tex]\( 5x + 3y^6 \)[/tex] results in [tex]\( 500 x^3 + 108 y^{18} \)[/tex], we conclude that [tex]\( 5x + 3y^6 \)[/tex] is not a factor.
### Checking if [tex]\( 25 x^2 + 15 x y^6 + 9 y^{12} \)[/tex] is a factor:
For [tex]\( 25 x^2 + 15 x y^6 + 9 y^{12} \)[/tex] to be a factor of [tex]\( 500 x^3 + 108 y^{18} \)[/tex], we need:
- When [tex]\( 500 x^3 + 108 y^{18} \)[/tex] is divided by [tex]\( 25 x^2 + 15 x y^6 + 9 y^{12} \)[/tex], it should yield a quotient that is in polynomial form without a remainder.
Given that the degrees of polynomial do not match without heavy manipulation and further verifying structure reformulation, it becomes evident upon structural analysis that [tex]\( 500 x^3 + 108 y^{18} \)[/tex] does not factorize into [tex]\( 25 x^2 + 15 x y^6 + 9 y^{12} \)[/tex], thus, [tex]\( 25 x^2 + 15 x y^6 + 9 y^{12} \)[/tex] is not a factor.
### Conclusion:
Since none of the given choices divide [tex]\( 500 x^3 + 108 y^{18} \)[/tex] perfectly, none of the following choices are factors of the polynomial [tex]\( 500 x^3 + 108 y^{18} \)[/tex]:
- 6
- [tex]\( 5x + 3y^6 \)[/tex]
- [tex]\( 25 x^2 + 15 x y^6 + 9 y^{12} \)[/tex]
So, the answer is: None of the above.
