Algebraic Analysis of Power Sums for Three Variables

Mathematical Problem Overview: Power Sums of Three Variables

• This problem involves a system of three non-linear equations with three variables ($a$, $b$, and $c$) and requires finding the value of a specific expression related to the fourth power of these variables, based on the patterns established by the first three equations.
• The given information is as follows:
  • Sum of the variables: $a + b + c = 4$
  • Sum of the squares: $a^2 + b^2 + c^2 = 10$
  • Sum of the cubes: $a^3 + b^3 + c^3 = 22$
• The objective is to determine the value of the fourth power sum: $a^4 + b^4 + c^4 = ?$

Theoretical Framework: Newton-Girard Formulas

• The most efficient method to solve for higher power sums without solving for individual variables ($a$, $b$, and $c$) is using the Newton-Girard Formulas.

• These formulas relate power sums ($p_n$) to elementary symmetric polynomials ($e_n$).

• Let:

  • $p_1 = a + b + c$
  • $p_2 = a^2 + b^2 + c^2$
  • $p_3 = a^3 + b^3 + c^3$
  • $p_k = a^k + b^k + c^k$

• Let the elementary symmetric polynomials be:

  • $e_1 = a + b + c$
  • $e_2 = ab + bc + ca$
  • $e_3 = abc$

• The recursive Newton-Girard relations for three variables are defined by:

  • $p_1 - e_1 = 0$
  • $p_2 - e_1p_1 + 2e_2 = 0$
  • $p_3 - e_1p_2 + e_2p_1 - 3e_3 = 0$
  • $p_n - e_1p_{n-1} + e_2p_{n-2} - e_3p_{n-3} = 0$ (for $n \ge 3$)

Step-by-Step Calculation of Elementary Symmetric Polynomials

• Step 1: Determine $e_1$

  • From the first equation and the definition of $e_1$:
  • $e_1 = a + b + c = 4$

• Step 2: Determine $e_2$

  • Using the relationship $p_2 = e_1p_1 - 2e_2$:
  • Substitute the known values: $10 = (4)(4) - 2e_2$
  • Simplify the equation: $10 = 16 - 2e_2$
  • Subtract 16 from both sides: $-6 = -2e_2$
  • Divide by -2: $e_2 = 3$

• Step 3: Determine $e_3$

  • Using the relationship $p_3 = e_1p_2 - e_2p_1 + 3e_3$:
  • Substitute the known values: $22 = (4)(10) - (3)(4) + 3e_3$
  • Simplify the equation: $22 = 40 - 12 + 3e_3$
  • Further simplification: $22 = 28 + 3e_3$
  • Subtract 28 from both sides: $-6 = 3e_3$
  • Divide by 3: $e_3 = -2$

Determination of the Fourth Power Sum

• To find $p_4 = a^4 + b^4 + c^4$, we utilize the recursive formula for $n = 4$:
  • $p_4 = e_1p_3 - e_2p_2 + e_3p_1$
• Substitute the calculated values for $e_1$, $e_2$, and $e_3$, along with the given values for $p_1$, $p_2$, and $p_3$:
  • $p_4 = (4)(22) - (3)(10) + (-2)(4)$
• Perform the multiplications:
  • $(4)(22) = 88$
  • $(3)(10) = 30$
  • $(-2)(4) = -8$
• Calculate the final result:
  • $p_4 = 88 - 30 - 8$
  • $p_4 = 58 - 8$
  • $p_4 = 50$

Summary of Results

• Based on the algebraic properties of symmetric polynomials:
  • The sum of the variables is $4$.
  • The sum of their pairwise products is $3$.
  • The product of the three variables is $-2$.
  • The value of the expression $a^4 + b^4 + c^4$ is exactly $50$.