Exercise 1830Algebra › PolynomialsL1Added 3 Nov 2026

Symmetric functions of roots

#symmetric functions#Vieta#Newton identities

Statement

Let $x,y,z$ be the roots of $X^3-X^2-2X+1=0$ . Compute $x^2+y^2+z^2$ and $x^3+y^3+z^3$ .

Hint

Vieta plus Newton's identities: $p_k=x^k+y^k+z^k$ satisfies $p_k=e_1p_{k-1}-e_2p_{k-2}+e_3p_{k-3}$ where $e_1,e_2,e_3$ are the elementary symmetric functions.

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Solution

Vieta. $e_1=1$ , $e_2=-2$ , $e_3=-1$ .

$p_2$ . $p_2=e_1^2-2e_2=1+4=5$ .

$p_3$ . $p_3=e_1p_2-e_2p_1+3e_3=1\cdot5-(-2)\cdot1+3\cdot(-1)=5+2-3=4$ .

Conclusion. $x^2+y^2+z^2=5$ and $x^3+y^3+z^3=4$ .

Symmetric functions of roots

Statement

Solution

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