Question

已知a+b+c=0,a的二次方+b二次方+c的二次方=4,试求a的四次方+b的四次方+c的四次方的

Answer 1

（ab+ac+bc)^2=(ab)^2+(ac)^2+(bc)^2+2a^2bc+2b^2ac+2c^2ab

=(ab)^2+(ac)^2+(bc)^2+2abc(a+b+c)

=(ab)^2+(ac)^2+(bc)^2

0=(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)=0

(ab+bc+ac)=-1/2

所以(ab)^2+(ac)^2+(bc)^2=1/4

1=(a^2+b^2+c^2)^2=a^4+b^4+c^4+2[(ab)^2+(ac)^2+(bc)^2

a^4+b^4+c^4=1-2*1/4=1/2