If a + b + c = 0, then show that

[agoogler]

Homework Statement

If a + b + c = 0, then show that
6(a^5 + b^5 + c^5) = 5(a^3 + b^3 + c^3)(a^2 + b^2 + c^2)

Homework Equations

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The Attempt at a Solution

Since a+b+c=0 then a^3 + b^3 + c^3=3abc .
But then what? I've no idea , Please help.

 

[Vacrin]

Im sorry to say, but none of these equations work, they are not equal so they have no solutions.

I did a quick test by putting numbers in for a b and c. Being 2,3,4 the first part 6(a^5+b^5+c^5) = 5(a^3+b^3+c^3)(a^2+b^2+c^2) using the numbers i offered earlier. It comes to 7794=14355.

The second equation comes to 99=72

 

[agoogler]

Im sorry to say, but none of these equations work, they are not equal so they have no solutions.

I did a quick test by putting numbers in for a b and c. Being 2,3,4 the first part 6(a^5+b^5+c^5) = 5(a^3+b^3+c^3)(a^2+b^2+c^2) using the numbers i offered earlier. It comes to 7794=14355.

The second equation comes to 99=72

Please read the question carefully. It says that the above holds only if a+b+c=0 and that is what I want to prove.

 

[Vacrin]

Well, that equation has quite a few solutions, wouldn't you need to solve for them all? Or does mater in the equations given below?

 

[agoogler]

Well, that equation has quite a few solutions, wouldn't you need to solve for them all? Or does mater in the equations given below?

I do not want to solve for solutions . ( By the way the solutions are all a , b and c that sum to 0) . But I want to prove that equation when a+b+c=0.

 

[Tsunoyukami]

I haven't tried this but the obvious thing would be to expand both sides and simplify by using the condition a + b + c = 0.

 

[Vacrin]

I do not want to solve for solutions . ( By the way the solutions are all a , b and c that sum to 0) . But I want to prove that equation when a+b+c=0.

No no, i mean, solve the other given equations using things like a+b=-c

 

[Ray Vickson]

Please read the question carefully. It says that the above holds only if a+b+c=0 and that is what I want to prove.

Put c = -a-b on both sides and expand both sides.

 

[agoogler]

Put c = -a-b on both sides and expand both sides.

Thanks a lot !
After expanding I get LHS = -30 a^4 b-60 a^3 b^2-60 a^2 b^3-30 a b^4
RHS= -30 a^4 b-60 a^3 b^2-60 a^2 b^3-30 a b^4
LHS=RHS
Q.E.D.