Final answer:
The expression a^3y + 1 cannot be factored over the integers as it does not fit the patterns for factoring such as common factors or being a sum/difference of cubes.
Step-by-step explanation:
The expression that needs to be factored is a^3y + 1. However, this expression cannot be factored over the integers since it is not a difference of cubes or a sum of cubes that would allow for a straightforward factoring process. The given expression is a binomial with no common factors other than 1.
Factoring typically involves expressing an algebraic expression as the product of two or more simpler expressions that, when multiplied together, yield the original expression.
To factor an expression completely, one would look for common factors, use identities like the difference of squares or the sum/difference of cubes, and consider if there is a possibility to apply the distributive property or any other factoring technique. In this case, no such method is applicable, and the expression stands as is.
When considering cubing of exponentials, the general rule is to cube the constant term and multiply the exponent of the variable by 3. For example, if one were to cube (2a)^3, the result would be 8a^9. Applying such rules depends on recognizing patterns in the expressions that match the factoring identities.