<s>[INST] Number of polynomials which are divisible by $x+1$ Let $a,b,c,d$ be four integers (not necessarily distinct) in the set ${1,2,3,4,5}$ . The number of polynomials $f(x)=x^4+ax^3+bx^2+cx+d$ which are divisible by $x+1$ are:
$(A)$ Between 55 and 65
$(B)$ Between 65 and 85
$(C)$ Between 86 and 105
$(D)$ More than 105
I see that it will be divisible when $f(-1)=0$
i.e. $1-a+b-c+d=0$ but how should I proceed further ?
 [/INST] Your approach of looking for the number of solutions to $1-a+b-c+d=0$ is excellent.
Let $f(n)$ be the number of ways of choosing two (possibly identical) numbers $h,k$ from $\{1,2,3,4,5\}$ so that $h-k=n$. It is easy to check that $f(n)=1,2,3,4,5,4,3,2,1$ for $n=-4,-3,-2,\dots,3,4$ and 0 for all other $n$.
So $1-a+b$ must be one of $-3,-2,\dots,4,5$. The number of $(a,b)$ giving each of these is $1,2,3,4,5,4,3,2,1$. The number of $(c,d)$ giving minus those values is $2,3,4,5,4,3,2,1,0$. So the total number of solutions is $1\cdot2+2\cdot3+3\cdot4+4\cdot5+5\cdot4+4\cdot3+3\cdot2+2\cdot1=80$.
So the answer to the multiple choice question is (B).
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