若a，b，c都是有理数，且a+b+c=0，a3+b3+c3=0，求代数式a5+b5+c5的值

Question

如题所述

Answer 1

∵a+b+c=0
∴c=-（a+b）
∵a³+b³+c³=0
∴ a³+b³+[-(a+b)]³=0
∴ a³+b³-a-b³-3a²b-3ab²=0
∴3a²b+3ab²=0
∴ab(a+b)=0

a^5+b^5+c^5
= a^5+b^5+[-(a+b)]^5
= a^5+b^5-a^5-5a^4b-10a^3b^2-10a^2b^3-5ab^4-b^5
= -5a^4b-10a^3b^2-10a^2b^3-5ab^4
= -5a^4b-5ab^4-10a^3b^2-10a^2b^3
= -5ab³(a+b) - 10a²b²(a+b)
= -5ab(a+b) (b²+2ab) = 0

Answer 2

供参考

Answer 3