Newton's Identities and Power Sum Calculation

Problem Identification and System Overview

• The provided transcript presents a system of polynomial equations involving three variables, $a$, $b$, and $c$, and seeks the value of a higher-order expression.
• The known values are as follows:
  • The first power sum ($p_1$): $a + b + c = 4$
  • The second power sum ($p_2$): $a^2 + b^2 + c^2 = 10$
  • The third power sum ($p_3$): $a^3 + b^3 + c^3 = 22$
• The target expression is listed as "a² + b + c 4 =?", which, in the context of power sum sequences, is interpreted as the fourth power sum ($p_4$):
  • Find the value of: $a^4 + b^4 + c^4 = ?$

Mathematical Framework: Newton's Identities

• To solve for the sum of higher powers, we use Newton's Identities (also known as the Newton-Girard formulae). These identities relate power sums ($p_k$) to elementary symmetric polynomials ($e_k$).
• Definition of Power Sums ($p_k$):
  • $p_1 = a + b + c$
  • $p_2 = a^2 + b^2 + c^2$
  • $p_3 = a^3 + b^3 + c^3$
  • $p_k = a^k + b^k + c^k$
• Definition of Elementary Symmetric Polynomials ($e_k$) for three variables:
  • $e_1 = a + b + c$
  • $e_2 = ab + bc + ca$
  • $e_3 = abc$
• General Recursive Formula:
  • For $k \le n$ (where $n$ is the number of variables):
  • $p_k - e_1 p_{k-1} + e_2 p_{k-2} - ( \text{...} ) + (-1)^{k-1} e_{k-1} p_1 + (-1)^k k e_k = 0$
  • For $k > n$:
  • $p_k - e_1 p_{k-1} + e_2 p_{k-2} - ( \text{...} ) + (-1)^n e_n p_{k-n} = 0$

Determining the Elementary Symmetric Polynomials

• In order to calculate $p_4$, we must first determine the values of $e_1$, $e_2$, and $e_3$.

• Step 1: Determine $e_1$

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

• Step 2: Determine $e_2$

  • We use the relationship between the first and second power sums:
  • $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$
  • Substituting the known values:
  • $(4)^2 = 10 + 2e_2$
  • $16 = 10 + 2e_2$
  • $6 = 2e_2$
  • $e_2 = 3$

• Step 3: Determine $e_3$

  • We use Newton's Identity for $k = 3$:
  • $p_3 - e_1 p_2 + e_2 p_1 - 3e_3 = 0$
  • Substitute the values: $p_3 = 22$, $e_1 = 4$, $p_2 = 10$, $e_2 = 3$, $p_1 = 4$
  • $22 - (4)(10) + (3)(4) - 3e_3 = 0$
  • $22 - 40 + 12 - 3e_3 = 0$
  • $-18 + 12 - 3e_3 = 0$
  • $-6 - 3e_3 = 0$
  • $3e_3 = -6$
  • $e_3 = -2$

Calculating the Fourth Power Sum ($p_4$)

• With the values of $e_1$, $e_2$, and $e_3$ established, we apply Newton's Identity for $k = 4$. Since there are only three variables ($n=3$), we use the formula for $k > n$:

  • $p_4 - e_1 p_3 + e_2 p_2 - e_3 p_1 = 0$

• Plug in the established values:

  • $e_1 = 4$
  • $e_2 = 3$
  • $e_3 = -2$
  • $p_3 = 22$
  • $p_2 = 10$
  • $p_1 = 4$

• The calculation:

  • $p_4 - (4)(22) + (3)(10) - (-2)(4) = 0$
  • $p_4 - 88 + 30 + 8 = 0$
  • $p_4 - 58 + 8 = 0$
  • $p_4 - 50 = 0$
  • $p_4 = 50$

Alternative Algebraic Verification

• One can verify this result using the algebraic expansion of $(a^2 + b^2 + c^2)^2$:

  • $(a^2 + b^2 + c^2)^2 = a^4 + b^4 + c^4 + 2(a^2 b^2 + b^2 c^2 + c^2 a^2)$
  • Substituting $p_2 = 10$:
  • $10^2 = p_4 + 2(a^2 b^2 + b^2 c^2 + c^2 a^2)$
  • $100 = p_4 + 2(a^2 b^2 + b^2 c^2 + c^2 a^2)$

• To find $a^2 b^2 + b^2 c^2 + c^2 a^2$, use the square of $e_2$:

  • $e_2^2 = (ab + bc + ca)^2 = a^2 b^2 + b^2 c^2 + c^2 a^2 + 2abc(a + b + c)$
  • Substitute values $e_2 = 3$, $e_3 = -2$, and $e_1 = 4$:
  • $3^2 = (a^2 b^2 + b^2 c^2 + c^2 a^2) + 2(-2)(4)$
  • $9 = (a^2 b^2 + b^2 c^2 + c^2 a^2) - 16$
  • $a^2 b^2 + b^2 c^2 + c^2 a^2 = 25$

• Now substitute this back into the $p_4$ verification equation:

  • $100 = p_4 + 2(25)$
  • $100 = p_4 + 50$
  • $p_4 = 50$

• Both methods confirm that the value for the fourth power sum is $50$.