Q: Derivation via the Binomial Square Identity

To determine the value of ##\alpha^2 + \beta^2 + \gamma^2## , we must bridge the gap between elementary symmetric polynomials and power sums. The most direct approach involves the expansion of the square of the first elementary symmetric polynomial. Mathematically, we utilize the multinomial expansion ###(\alpha + \beta + \gamma)^2 = \alpha^2 + \beta^2 + \gamma^2 + 2(\alpha\beta + \beta\gamma + \gamma\alpha)###. This identity is a fundamental tool in algebraic manipulation, revealing that the square of the sum of roots is composed of the sum of their individual squares and twice the sum of their pairwise products.

Q: Application to Higher-Order Symmetric Sums

While our primary task focused on ##S_2## , the same technical framework allows us to calculate ##S_3 = \alpha^3 + \beta^3 + \gamma^3## with minimal extra effort. Using the recurrence relation established previously, we can write ###S_3 = 5S_2 - 2S_1 + 8S_0###. It is crucial to remember that ##S_0## is the sum of the roots to the power of zero, which is ###1 + 1 + 1 = 3### for a cubic. Substituting our known values, we get ###S_3 = 5(21) - 2(5) + 8(3) = 105 - 10 + 24 = 119###. This rapid calculation showcases the utility of the symmetric sum methodology in handling higher-order terms.

Question 1

Defining the Cubic Equation and Roots

Accepted Answer

The mathematical exploration of polynomial roots begins with the formal definition of the cubic equation presented in the problem statement. Specifically, we are tasked with analyzing the monic polynomial f(x) = x^3 - 5x^2 + 2x - 8 = 0. In this technical context, the term monic refers to the leading coefficient being unity, which significantly simplifies the application of algebraic identities. By identifying the coefficients of the general form ###ax^3 + bx^2 + cx + d = 0###, we establish that a = 1, b = -5, c = 2, and d = -8. These values are essential parameters for the subsequent implementation of Viete's Formulas and Symmetric Sums, as they dictate the relationships between the roots and the polynomial coefficients.

Question 2

Establishing Fundamental Relations

Accepted Answer

The first fundamental relation derived from Vieta’s work concerns the sum of the roots taken one at a time. This is defined as the first elementary symmetric polynomial, denoted as ##e_1## or ##\sigma_1##. In terms of the coefficients of a cubic equation, this sum is equal to the negative of the coefficient of the squared term divided by the leading coefficient. Applying this to our equation, we obtain ###\alpha + \beta + \gamma = -\frac{b}{a} = -\frac{-5}{1} = 5###. This value represents the trace of the companion matrix associated with the polynomial and serves as the primary linear constraint on the root distribution.

Question 3

Derivation via the Binomial Square Identity

Accepted Answer

To determine the value of ##\alpha^2 + \beta^2 + \gamma^2##, we must bridge the gap between elementary symmetric polynomials and power sums. The most direct approach involves the expansion of the square of the first elementary symmetric polynomial. Mathematically, we utilize the multinomial expansion ###(\alpha + \beta + \gamma)^2 = \alpha^2 + \beta^2 + \gamma^2 + 2(\alpha\beta + \beta\gamma + \gamma\alpha)###. This identity is a fundamental tool in algebraic manipulation, revealing that the square of the sum of roots is composed of the sum of their individual squares and twice the sum of their pairwise products.

Question 4

Technical Execution and Result Verification

Accepted Answer

With the derivation complete, we proceed to substitute the previously established values into the sum of squares formula. We recall that the sum of the roots is ##\alpha + \beta + \gamma = 5## and the pairwise product sum is ##\alpha\beta + \beta\gamma + \gamma\alpha = 2##. Plugging these into our equation, we compute the square of the first term ###(5)^2 = 25### and twice the second term ###2(2) = 4###. The final calculation proceeds as ###25 - 4 = 21###. Therefore, the sum of the squares of the roots for the cubic equation x^3 - 5x^2 + 2x - 8 = 0 is exactly 21.

Question 5

Implementing Newton-Girard Identities

Accepted Answer

For advanced problem solving, one can utilize the Newton-Girard identities, which provide a recursive relation between power sums and elementary symmetric polynomials. Let ##S_k = \alpha^k + \beta^k + \gamma^k## be the k-th power sum. For a cubic polynomial with coefficients a, b, c, d, the identities state that ###aS_1 + b = 0### and ###aS_2 + bS_1 + 2c = 0###. Given a = 1 and b = -5, the first identity yields ###S_1 - 5 = 0 \implies S_1 = 5###. The second identity then becomes ###S_2 - 5(5) + 2(2) = 0###, which simplifies to ###S_2 - 25 + 4 = 0###, resulting in ##S_2 = 21##.

Question 6

Application to Higher-Order Symmetric Sums

Accepted Answer

While our primary task focused on ##S_2##, the same technical framework allows us to calculate ##S_3 = \alpha^3 + \beta^3 + \gamma^3## with minimal extra effort. Using the recurrence relation established previously, we can write ###S_3 = 5S_2 - 2S_1 + 8S_0###. It is crucial to remember that ##S_0## is the sum of the roots to the power of zero, which is ###1 + 1 + 1 = 3### for a cubic. Substituting our known values, we get ###S_3 = 5(21) - 2(5) + 8(3) = 105 - 10 + 24 = 119###. This rapid calculation showcases the utility of the symmetric sum methodology in handling higher-order terms.

Question 7

Structural Properties of Symmetric Polynomials

Accepted Answer

The theoretical foundation of this problem lies in the Fundamental Theorem of Symmetric Polynomials. This theorem asserts that any polynomial in n variables that is invariant under permutations of those variables can be expressed as a unique polynomial in the elementary symmetric polynomials ##e_1, e_2, \dots, e_n##. This structural property is what allowed us to calculate ##\alpha^2 + \beta^2 + \gamma^2## as ##e_1^2 - 2e_2##. This theorem is a cornerstone of ring theory and algebraic geometry, providing a link between the geometry of root spaces and the algebra of coefficient rings.

Question 8

The Legacy of François Viète and Albert Girard

Accepted Answer

The formulas we use today are named after François Viète, a 16th-century French mathematician often called the father of modern algebraic notation. Viète was the first to use letters to represent both known and unknown quantities, a revolutionary step that allowed for the general formulation of polynomial relations. However, Viète primarily worked with positive real roots, as the concept of negative and complex numbers was still maturing. His work in In artem analyticam isagoge laid the groundwork for the systematic study of Viete's Formulas and Symmetric Sums that we rely on in contemporary mathematics.

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