Question

已知a+b+c+d=o,a的立方+b的立方+c的立方+d的立方=3,求abc+bcd+cda+dab的值

Answer 1

a+b+c+d＝0 =>a+b=-(c+d)--------两边3次方 =>(a+b)^3=-(c+d)^3 =>(a+b)^3+(c+d)^3=0 =>a^3+b^3+c^3+d^3+3(a^2b+b^2a+c^2d+d^2c)=0 =>a^3+b^3+c^3+d^3+3[ab(a+b)+cd(c+d)]=0----a+b=-(c+d); =>a^3+b^3+c^3+d^3=3[ab(c+d)+cd(a+b)]----展开就是结论 =>a^3+b^3+c^3+d^3=3(abc+bcd+cda+abd)=3 abc+bcd+cda+dab的值:1