Exercise
Which of the following pairs of polynomials in Q[X] is not congruent modulo d = 1 + 2X?
1 + X and -X.
1 + X2 + 2X5 and 1 + 3X4.
4X2 and 1.
1 and 2X.
These two polynomials are congruent modulo d, because their difference
is divisible by d.
These two
polynomials are congruent modulo d, because their difference
X2 - 3X4 + 2X5
= X2(1 + 2X)(1 - 2X +
X2) is divisible by d.
These two polynomials are congruent modulo d, because their difference
-1 + 4X2 = (-1 + 2X)(1 + 2X) is divisible
by d.
Indeed, these polynomials are not congruent modulo d, because their
difference 1 - 2X is not divisible by d.