Let α, β, γbe the roots of the equation 3 x^3-x^2-3 x+1=0 . If α, β, γare in H.P. then |α-γ|= (A

step 1 So in this question let alpha is equal to 1 by a minus t, beta is equal to 1 by a and comma is e

Let α, β, γbe the roots of the equation 3 x^3-x^2-3 x+1=0 ....

Let $\alpha, \beta, \gamma$ be the roots of the equation $3 x^{3}-x^{2}-3 x+1=0 .$ If $\alpha, \beta, \gamma$ are in H.P. then $|\alpha-\gamma|=$
(A) $\frac{1}{3}$
(B) $\frac{2}{3}$
(C) $\frac{4}{3}$
(D) None of these

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Key Concepts

Polynomial Equations

Understanding polynomial equations involves recognizing how the roots interact with the coefficients through structured relationships. This is particularly useful when additional conditions, such as the roots forming a specific progression, are imposed on the equation.

Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to the sums and products of its roots. These formulas are crucial in problems involving polynomial equations as they provide algebraic relationships among the roots without requiring explicit calculation of each root.

Harmonic Progression

A harmonic progression (H.P.) is a sequence in which the reciprocals of its terms form an arithmetic progression (A.P.). This concept is essential for converting problems involving H.P. into ones involving A.P., making it easier to analyze relationships among terms.

Arithmetic Progression

An arithmetic progression is a sequence of numbers with a constant difference between successive terms. This concept is used as the foundational structure when dealing with reciprocal transformations in harmonic progressions.

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