Symmetric Polynomial System Problem

Vocabulary flashcards detailing the symmetric polynomial system, relevant elementary symmetric identities, and the computed value for the fourth power sum.

System of Equations

A set of symmetric power sums given as $a + b + c = 4$, $a^2 + b^2 + c^2 = 10$, and $a^3 + b^3 + c^3 = 22$, used to find the value of $a^4 + b^4 + c^4$.

Elementary Symmetric Sum $ab + bc + ca$

The second elementary symmetric polynomial $e_2 = ab + bc + ca$, which evaluates to $3$ using the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$.

Elementary Symmetric Product $abc$

The product $e_3 = abc$, which evaluates to $-2$ using Newton's sums relation $p_3 - e_1 p_2 + e_2 p_1 - 3 e_3 = 0$.

Fourth Power Sum $a^4 + b^4 + c^4$

The fourth power sum $p_4 = a^4 + b^4 + c^4$, which equals $50$ using the recurrence relation $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$.