Let a, b, c, d be four integers (not necessarily distinct) in the set {1, 2, 3, 4, 5}. the number of polynomials x4 + ax3 + bx2 + cx + d which is divisible by x + 1 is(
a. between 55 and 65. (
b. between 66 and 85.(
c. between 86 and 105. (
d. more than 105.

By the polynomial remainder theorem, will be a factor of if the remainder upon division is 0, and this remainder is given by :





Since , it follows that . But notice that if , then we have



and since , the equation above requires that either or , which is impossible. So .

So we have 8 cases to check:

(1) Notice that if , we have . This is only possible for .

(2) If , then , and so we can have .

(3) If , then , and so .

(4) If , then .

(5) If , then .

(6) If , then .

(7) If , then .

(8) If , then .

At the same time, we have 8 cases to consider to find how many options there are for .

(1) . We have only one choice of .

(2) . This is the same as when , which we found to be 2 choices.

(3) Same as ; 3 choices.

(4) Same as ; 4 choices.

(5) 5.

(6) 4.

(7) 3.

(8) 2.

In total, there are






ways to choose such that is a factor of , so the answer is B.

Note the symmetry of the sum above. You can easily give a slightly briefer combinatorial argument for this answer, but I figured a more brute-force approach would be easier to follow.

The answer to this question is 3