algebraic problem

**alexandros** · June 23rd 2010, 03:57 PM

Challenge problem:

Express the following algebraic quantities in terms of the variables λ,μ :

1) $a^2+b^2+c^2$

2) $a^3+b^3+c^3$

3) $a^5+b^5+c^5$

Where $a\neq b\neq c$ are the roots of the following equation:

$x^3+\lambda x+\mu=0$ , $\lambda\neq 0,\mu\neq 0$ and real Nos

Moderator edit: Approved Challenge question.

**simplependulum** · June 23rd 2010, 10:20 PM

(a) $a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+bc+ca) = -2A$

(b) we have $a^3 + A a + B = 0$

$a^3 = -Aa - B$

$a^3 + b^3 + c^3 = -A(a+b+c) - 3B = -3B$

(c) $a^5 + Aa^3 + Ba^2 = 0$

$a^5 = -Aa^3 - Ba^2$

$a^5 + b^5 + c^5 = -A(a^3 + b^3 + c^3) - B ( a^2 + b^2 + c^2) = 3AB + 2AB = 5AB$

**alexandros** · June 24th 2010, 07:06 AM

Originally Posted by simplependulum

(a) $a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+bc+ca) = -2A$

(b) we have $a^3 + A a + B = 0$

$a^3 = -Aa - B$

$a^3 + b^3 + c^3 = -A(a+b+c) - 3B = -3B$

(c) $a^5 + Aa^3 + Ba^2 = 0$

$a^5 = -Aa^3 - Ba^2$

$a^5 + b^5 + c^5 = -A(a^3 + b^3 + c^3) - B ( a^2 + b^2 + c^2) = 3AB + 2AB = 5AB$

O.K correct ,but why don't you show how you get that :

a+b+c=0 and ab+ac+bc = 2A ? Not for me but for other people that want to know the solution of the problem?

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