Symmetric Polynomials and Power Sum Derivations Notes

Initial System of Equations

The mathematical transcript provides a set of three equations involving three variables ($a$, $b$, and $c$) and a final expression to be evaluated. The given values are as follows:

• First Order Sum: $a + b + c = 4$
• Second Order Sum (Power of 2): $a^2 + b^2 + c^2 = 10$
• Third Order Sum (Power of 3): $a^3 + b^3 + c^3 = 22$
• Target Expression: The transcript lists the final query as "a² + b + c 4 =?". In the context of the mathematical progression shown ($n=1, 2, 3$), this is systematically interpreted as the calculation of the sum of the fourth powers: $a^4 + b^4 + c^4$.

Theoretical Background: Symmetric Polynomials and Newton-Girard Sums

To solve for higher-order power sums when the lower-order sums are known, we utilize the theory of Symmetric Polynomials. Specifically, the Newton-Girard identities provide a direct recursive relationship between power sums and elementary symmetric polynomials.

Definitions of Elementary Symmetric Polynomials (for 3 Variables)

• First Elementary Symmetric Polynomial ($e_1$):     $e_1 = a + b + c$
• Second Elementary Symmetric Polynomial ($e_2$):     $e_2 = ab + bc + ca$
• Third Elementary Symmetric Polynomial ($e_3$):     $e_3 = abc$

Definition of Power Sums ($p_n$)

• $p_1 = a^1 + b^1 + c^1$
• $p_2 = a^2 + b^2 + c^2$
• $p_3 = a^3 + b^3 + c^3$
• $p_4 = a^4 + b^4 + c^4$

Calculation of Elementary Symmetric Polynomials

Before determining the fourth power sum, we must find the values of $e_1$, $e_2$, and $e_3$ based on the provided data.

Calculating $e_1$

From the first given equation: $e_1 = a + b + c = 4$

Calculating $e_2$

We use the relationship between $p_2$, $e_1$, and $e_2$: $p_2 = e_1 p_1 - 2e_2$ Substituting the known values ($p_2 = 10$, $p_1 = 4$, $e_1 = 4$): $10 = (4)(4) - 2e_2$$10 = 16 - 2e_2$$-6 = -2e_2$$e_2 = 3$

Calculating $e_3$

We use the Newton-Girard identity for the third power sum: $p_3 = e_1 p_2 - e_2 p_1 + 3e_3$ Substituting the known values ($p_3 = 22$, $p_2 = 10$, $p_1 = 4$, $e_1 = 4$, $e_2 = 3$): $22 = (4)(10) - (3)(4) + 3e_3$$22 = 40 - 12 + 3e_3$$22 = 28 + 3e_3$$-6 = 3e_3$$e_3 = -2$

Derivation of the Fourth Power Sum

The target expression corresponds to $p_4$. For a system of three variables ($a, b, c$), the elementary symmetric polynomial $e_n = 0$ for all $n > 3$. The recursive formula for $p_4$ is:

$p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$

Step-by-Step Numerical Substitution

1. Identify Constants:
   • $e_1 = 4$
   • $e_2 = 3$
   • $e_3 = -2$
2. Identify Power Sums:
   • $p_1 = 4$
   • $p_2 = 10$
   • $p_3 = 22$
3. Execute Calculation:
   • $p_4 = (4)(22) - (3)(10) + (-2)(4)$
   • $p_4 = 88 - 30 - 8$
   • $p_4 = 58 - 8$
   • $p_4 = 50$

Summary of Final Results

By applying algebraic identities and Newton's Sums to the provided transcript data, the following values are derived:

• Elementary Sum ($e_1$): $4$
• Pairwise Sum ($e_2$): $3$
• Product of Variables ($e_3$): $-2$
• Sum of Fourth Powers ($p_4$): $50$

The answer to the query "a² + b + c 4 =?" (interpreted as $a^4 + b^4 + c^4$) is 50.