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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?

 Jun 9, 2019
 #1
avatar+33659 
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As follows:

 

 Jun 9, 2019
 #2
avatar+26397 
+3

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))=02(x3+y3+z3)+6xyz=06xyz=2(x3+y3+z3)xyz=13(x3+y3+z3)xyz=πe3

 

laugh

 Jun 9, 2019

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