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

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06-05-2017
Mathematics

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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

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LammettHash LammettHash · Answer 1 · 2017-05-08T17:49:32+02:00

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

[tex]\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}[/tex]

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

[tex]a-b+c-d-1=0[/tex]

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.