The sum of 3 real numbers is known to be zero. If the sum of their cubes is πe, what is their product equal to?
The sum of 3 real numbers is known to be zero.
If the sum of their cubes is πe, what is their product equal to?
x+y+z=0x3+y3+z3=πe
1.)
0=(x+y+z)3=x3+y3+z3+6xyz+3(x2(y+z)+y2(x+z)+z2(x+y))
2.)
0=(x+y+z)(x2+y2+z2)=x3+y3+z3+x2(y+z)+y2(x+z)+z2(x+y)x2(y+z)+y2(x+z)+z2(x+y)=−(x3+y3+z3)
x3+y3+z3+6xyz+3(x2(y+z)+y2(x+z)+z2(x+y))=0x3+y3+z3+6xyz+3(−(x3+y3+z3))=0−2(x3+y3+z3)+6xyz=06xyz=2(x3+y3+z3)xyz=13(x3+y3+z3)xyz=πe3