Solving Systems of Power Sum Equations for Three Variables

Problem Statement and Identified Goal

• The transcript identifies a system of three nonlinear equations containing three variables: $a$, $b$, and $c$.
• The primary objective is to find the value of a certain expression based on the given powers of the variables.
• Given Equations:
  • Equation 1: $a + b + c = 4$
  • Equation 2: $a^2 + b^2 + c^2 = 10$
  • Equation 3: $a^3 + b^3 + c^3 = 22$
• Target Expression: The transcript lists "a² + b + c 4 =?". Given the progression of powers in the previous lines (1st, 2nd, and 3rd powers), this is interpreted as a request for the sum of the fourth powers: $a^4 + b^4 + c^4$.

Elementary Symmetric Polynomials of the Variables $a$, $b$, and $c$

• To solve for higher power sums, we utilize the elementary symmetric polynomials for three variables, defined as:
  • $e_1 = a + b + c$
  • $e_2 = ab + bc + ca$
  • $e_3 = abc$
• From the first given equation, we immediately establish:
  • $e_1 = 4$
• The higher-order polynomials $e_2$ and $e_3$ must be derived from the second and third power sums provided in the transcript.

Newton's Identities and Recurrence Relations

• Newton's Identities provide a relationship between power sums $p_k = a^k + b^k + c^k$ and elementary symmetric polynomials $e_k$.
• The recursive formulas for three variables are as follows:
  • $p_1 - e_1 = 0$
  • $p_2 - e_1 p_1 + 2e_2 = 0$
  • $p_3 - e_1 p_2 + e_2 p_1 - 3e_3 = 0$
  • $p_4 - e_1 p_3 + e_2 p_2 - e_3 p_1 + 4e_4 = 0$
• Note: For a system of three variables, $e_k = 0$ for all $k > 3$. Therefore, $e_4 = 0$, simplifying the equation for $p_4$.

Step-by-Step Derivation of Symmetric Polynomial Values

• Calculation of $e_2$:
  • Using the identity associated with $p_2$:
  • $10 - (4 \times 4) + 2e_2 = 0$
  • $10 - 16 + 2e_2 = 0$
  • $-6 + 2e_2 = 0$
  • $2e_2 = 6$
  • $e_2 = 3$
• Verification of $e_2$ using standard expansion:
  • $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$
  • $4^2 = 10 + 2e_2$
  • $16 = 10 + 2e_2 \rightarrow 6 = 2e_2 \rightarrow e_2 = 3$
• Calculation of $e_3$:
  • Using the identity associated with $p_3$:
  • $22 - (4 \times 10) + (3 \times 4) - 3e_3 = 0$
  • $22 - 40 + 12 - 3e_3 = 0$
  • $-18 + 12 - 3e_3 = 0$
  • $-6 - 3e_3 = 0$
  • $3e_3 = -6$
  • $e_3 = -2$
• Verification of $e_3$ using the cubic sum formula:
  • $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - (ab + bc + ca))$
  • $22 - 3e_3 = (4)(10 - 3)$
  • $22 - 3e_3 = 4 \times 7$
  • $22 - 3e_3 = 28$
  • $-3e_3 = 6 \rightarrow e_3 = -2$

Calculation of the Fourth Power Sum $a^4 + b^4 + c^4$

• With $e_1 = 4$, $e_2 = 3$, and $e_3 = -2$, we can now solve for $p_4 = a^4 + b^4 + c^4$ using the fourth Newton Identity where $e_4 = 0$:
  • $p_4 - e_1 p_3 + e_2 p_2 - e_3 p_1 = 0$
  • Substitute the known values into the equation:
  • $p_4 - (4 \times 22) + (3 \times 10) - (-2 \times 4) = 0$
• Perform the arithmetic operations:
  • $p_4 - 88 + 30 + 8 = 0$
  • $p_4 - 58 + 8 = 0$
  • $p_4 - 50 = 0$
• Solve for $p_4$:
  • $p_4 = 50$

Summary of Calculated Results

• The elementary symmetric polynomials for the set $\{a, b, c\}$ are:
  • Sum ($e_1$): $4$
  • Sum of pairwise products ($e_2$): $3$
  • Product of all variables ($e_3$): $-2$
• These values correspond to the coefficients of the characteristic cubic equation $x^3 - 4x^2 + 3x + 2 = 0$, whose roots are the values of $a$, $b$, and $c$.
• The value of the fourth power sum is determined to be:
  • $a^4 + b^4 + c^4 = 50$