Problem from: AoPS, Intermediate Algebra, Integer/Rational Roots

Question

for some integer m, the polynomial x^3 - 2011x + m has the three integer roots a, b and c. Find |a|+|b|+|c|

Answer 1

(x-a)(x-b)(x-c)

= (x^2 -2(a+b)x +ab)(x-c)

=x^3 -2(a+b)x^2 -cx^2 +abx -2c(a+b)x -abc

-2a-2b-c = 0, c=-2(a+b)

ab -2ac-2bc = -2011

2011= 2c(a+b)-ab

-abc = m

that's a start

3 equations 4 unknowns,

4th restriction is a,b,c are integers

as well as integer factors of m

by descartes, either 1 + root and 2 -, or 2+ and 1-

2011=-ab-4(a+b)^2

both a and b are odd integers, opposite signs

not sure there is a solution

17 and 5 could be close, if revising the problem

17(5)+4(17+5)^2 = 85+1936=2021

rewrite the problem with 2021, instead of -2011,

maybe it was a typo or misprint

then c=-44, m=-85(44)

|a|+|b|+|c|=17+5+44=66