Final answer:
To find roots of the polynomial x³ + x + 1 modulo 27, one must test integer values from 0 to 26 in the polynomial and find which values yield multiples of 27, thus indicating they are roots.
Step-by-step explanation:
To find all roots of the polynomial x³ + x + 1 modulo 27, we're looking for all values of x such that when x is substituted into the polynomial, the result is divisible by 27. The process involves trying different values of x in the range from 0 to 26 (since we're looking at modulo 27) and testing whether the polynomial evaluates to a multiple of 27.
Cubing of exponentials means to cube the digit term and multiply the exponent by 3. For example, if we have (3x)^3, the digit term 3 is cubed to become 27, and if there was an exponential term like x, its exponent would also be cubed. However, in this problem, we are not dealing with exponentials but rather a polynomial to be evaluated at integer values modulo 27.
Our procedure will involve computing x³ + x + 1 for each integer x between 0 and 26 and determining which, if any, give us a result that is a multiple of 27, indicating a root of the polynomial modulo 27.