2015/007) $(x-1)^3$ is a factor of $P(x) = x^{10}+ax^2 + bx + c$ find a + 2b + 3c

$(x-1)^3$ is a factor of $P(x) = x^{10}+ax^2 + bx + c$

so $x^3$ is a factor of $P(x+1) = (x+1)^{10}+ a(x+1)^2+b(x+1) + c$

so in the above expression coefficient of $x^2$, x and constant term shall be zero

coefficient of $x^2$ = $\binom{10}{2}+ a = 0= 45 + a = 0 $

coefficient of $x$ = 10 + 2a + b = 0

constant term = 1 + a + b + c = 0

solving we get a = - 45, b = 80, c = -36

so a + 2b + 3c = 7


This I have solved at http://mathhelpboards.com/challenge-questions-puzzles-28/find-2b-3c-14031.html#post66646. There are 2 more solutions