Question

let a,b,c,d be four integers (not necessarily distinct) in the set {1,2,3,4,5}. the no. of polynomials x^4+ax^3+bx^2+cx+d which is divisible by x+1 is

Answer 1

By either long or synthetic division, it's easy to show that

$\dfrac{x^4+ax^3+bx^2+cx+d}{x+1}=x^3+(a-1)x^2+(-a+b+1)x+(a-b+c-1)-\dfrac{a-b+c-d-1}{x+1}$

The quartic will be exactly divisible by $x+1$ when the numerator of the remainder term vanishes, or for those values of $a,b,c,d$ such that

$a-b+c-d-1=0$

I'm not sure how to count the number of solutions (software tells me it should be 80), but hopefully this is a helpful push in the right direction.