The real numbers x1, x2, x3 satisfying the equation x^3-x^2+b x+γ=0 are in A.P. The intervals in

step 1 Here, since x1, x2, and x3 are in arithmetic progression, so we will assume x1 is equal to alpha

The real numbers x1, x2, x3 satisfying the equation...

The real numbers $x_{1}, x_{2}, x_{3}$ satisfying the equation $x^{3}-x^{2}+b x+\gamma=0$ are in A.P. The intervals in which $\beta$ and $\gamma$ lie are (A) $\beta \in\left(-\infty, \frac{1}{3}\right]$ (B) $\beta \in\left[-\frac{1}{27}, \infty\right)$ (C) $\gamma \in\left(-\infty, \frac{1}{3}\right]$ (D) $\gamma \in\left[-\frac{1}{27}, \infty\right)$

The real numbers $x_{1}, x_{2}, x_{3}$ satisfying the equation $x^{3}-x^{2}+b x+\gamma=0$ are in A.P. The intervals in which $\beta$ and $\gamma$ lie are
(A) $\beta \in\left(-\infty, \frac{1}{3}\right]$
(B) $\beta \in\left[-\frac{1}{27}, \infty\right)$
(C) $\gamma \in\left(-\infty, \frac{1}{3}\right]$
(D) $\gamma \in\left[-\frac{1}{27}, \infty\right)$

Key Concepts

Cubic Polynomials and Real Root Conditions

A cubic polynomial is a degree-three polynomial that can have either one or three real roots. To ensure that all three roots are real, its discriminant must be non-negative, and additional relationships among the coefficients—derived from conditions like the roots forming an arithmetic progression—must be satisfied. This helps in determining the intervals in which the coefficients can lie.

Parameter Constraints from Root Structure

When the roots of a cubic polynomial are restricted to an arithmetic progression, the relationships provided by Viète's formulas yield specific equations linking the coefficients. Solving these equations typically leads to inequalities that the coefficients must satisfy in order for the polynomial to have three real roots in the desired configuration. This analysis is crucial for identifying the exact intervals for the parameters in question.

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence where the difference between any two successive terms is constant. In the context of polynomial roots, if the roots are in A.P., they can be expressed in terms of a common central term and a common difference. This structure simplifies the relationships among the roots, allowing us to express the sum, product, and other symmetric functions of the roots in terms of just two parameters.

Viète's Formulas

Viète's formulas provide a direct connection between the coefficients of a polynomial and the sums and products of its roots. In a cubic polynomial, these formulas relate the sum of the roots, the sum of the product of the roots taken two at a time, and the product of the roots to the coefficients of the polynomial. These relationships become particularly useful when there is a known pattern, such as an arithmetic progression, among the roots, as they help deduce constraints on the coefficients.

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