If α, β, γare the roots of x^3+a x^2+b x+c=0 . Then trace of...

If $\alpha, \beta, \gamma$ are the roots of $x^{3}+a x^{2}+b x+c=0 .$ Then trace of the matrix $\left|\begin{array}{lll}\alpha^{2} & \beta^{2} & \gamma^{2} \\ \gamma^{2} & \beta^{2} & \alpha^{2} \\ \beta^{2} & \alpha^{2} & \gamma^{2}\end{array}\right|$ is (a) a $^{2}$ (b) $a^{2}+2 b$ (c) $a^{2}-2 b$ (d) $a^{2}+b c$

If $\alpha, \beta, \gamma$ are the roots of $x^{3}+a x^{2}+b x+c=0 .$ Then trace of the matrix $\left|\begin{array}{lll}\alpha^{2} & \beta^{2} & \gamma^{2} \\ \gamma^{2} & \beta^{2} & \alpha^{2} \\ \beta^{2} & \alpha^{2} & \gamma^{2}\end{array}\right|$ is
(a) a $^{2}$
(b) $a^{2}+2 b$
(c) $a^{2}-2 b$
(d) $a^{2}+b c$

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Key Concepts

Viète’s Formulas

Viète’s formulas link the coefficients of a polynomial to symmetric sums of its roots. For example, in a cubic polynomial, the sum of the roots and the sum of the products of the roots taken two at a time are directly expressed in terms of the coefficients. This link is crucial for rewriting expressions involving the roots in terms of known quantities from the polynomial.

Symmetric Functions of Roots

Expressions like the sum of the squares of the roots are symmetric functions that can be decomposed into combinations of the elementary symmetric sums provided by Viète’s formulas. Such decompositions allow one to determine values of complex expressions solely based on the coefficients of the polynomial.

Matrix Trace

The trace of a matrix is the sum of its diagonal entries and is a fundamental concept in linear algebra. It often encapsulates key information about the matrix and, in problems involving roots of equations, facilitates the translation of matrix characteristics into algebraic expressions related to the roots.

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