To find the remainder when the polynomial 3x^4 + 2x^3 + x^2 + 2x + 14 is divided by 2, we can evaluate the polynomial at the root of the divisor. In this case, since we are dividing by 2, we will consider the polynomial at x = 0.
Substituting x = 0 into the polynomial:
3(0)^4 + 2(0)^3 + (0)^2 + 2(0) + 14 = 0 + 0 + 0 + 0 + 14 = 14
Now, we need to find the remainder of 14 when divided by 2. Performing the division:
14 ÷ 2 = 7 with a remainder of 0.
Thus, the remainder when 3x^4 + 2x^3 + x^2 + 2x + 14 is divided by 2 is 0.