If the polynomial x5 − 105 can be split as the product of the polynomials
x − 10 and a, what is a?

To answer this item, all that needs to be done is dividing the mathematical expression of x⁵ - 10⁵, our dividend, by x - 10, which is our divisor. 

The mathematical expression that can be used in order to properly express the given scenario is,
                        x⁵ - 10⁵ = (x - 10)(a)

The value of a, as stated above, can be calculated by the equation,
                     a = (x⁵ -10⁵) / (x - 10)

There are two ways to solve the problem: synthetic division and the long way. Each of which will yield the same answer. Simplifying the value of a will give us a final answer of,
      a = x⁴ + 10x³ + 100x² + 1000x