Is Newton Sums Relevant?

A degree 55 monic polynomial P(x)P(x) has 55 (not necessarily distinct) integer roots. Given that P(0)=2014P(0)=2014, how many ordered quintuplets of roots (r1,r2,r3,r4,r5)(r_1,r_2,r_3,r_4,r_5) satisfy the property that r1+r2+r3+r4+r5r_1+r_2+r_3+r_4+r_5 has the same units digit as r15+r25+r35+r45+r55r_1^5+r_2^5+r_3^5+r_4^5+r_5^5

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