For the cubic equation a x^3+b x^2+c x+d=0 the sum of the...

For the cubic equation $a x^{3}+b x^{2}+c x+d=0$ the sum of the squares of the roots is
(a) $\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}$
(b) $\frac{\mathrm{b}^{2}-2 \mathrm{ac}}{\mathrm{a}^{2}}$
(c) $\frac{b^{2}+2 a c}{a^{2}}$
(d) $(a+b+c)^{2}$

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Key Concepts

Symmetric Functions of Roots

Symmetric functions are expressions in the roots that remain unchanged under any permutation of the roots. For a cubic polynomial, common symmetric functions include the sum of the roots and the sum of the products of the roots taken two at a time. These symmetric expressions can be directly related to the coefficients using Vieta's formulas, offering a powerful tool for understanding root behavior.

Algebraic Identity for the Sum of Squares

The identity that expresses the sum of the squares of the roots as the square of the sum of the roots minus twice the sum of the products of the roots taken two at a time is essential. This identity, (sum of roots)^2 - 2(sum of pairwise products), bridges the gap between simple symmetric sums and more complex expressions like the sum of squares, enabling one to deduce the desired quantity using the relationships provided by Vieta's formulas.

Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. In the context of a cubic equation, these formulas provide expressions for the sum of the roots, the sum of the products of the roots taken two at a time, and the product of the roots, all in terms of the coefficients. This connection allows one to analyze the properties of the roots without explicitly solving the polynomial.

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