To find the remainder when polynomial P(x) = x⁵¹ + 51 is divided by (x + 1), we use the Remainder Theorem. This theorem states that the remainder of P(x) divided by (x – a) is P(a). Here, the divisor is (x + 1), which is (x – (– 1)), so a = – 1.

Therefore, we compute P(– 1):

P(− 1) = (− 1)51 + 51

Since (– 1) raised to any odd power is – 1, we get:

− 1 + 51 = 50

Thus, the remainder is 50.

Final answer: 50