If roots of x^3+h x^2+c x+d=0 are (a) in A.P. then 2 b^3-9 b c+27 d=0 (b) in G.P. then b^3 d=c^3

step 1 Hi, the question is given as if the roots of x q plus b x squared plus c x equal to 0 are in ap

If roots of x^3+h x^2+c x+d=0 are (a) in A.P. then 2 b^3-9 b...

Key Concepts

Geometric Progression

A geometric progression is a sequence where each term is derived by multiplying the previous term by a constant factor, known as the common ratio. For polynomial roots in geometric progression, the multiplicative structure sets up a distinct pattern which results in relationships between the coefficients. This is typically analyzed by expressing the roots in terms of an initial term and the common ratio, and then applying Viète’s formulas.

Repeated (Equal) Roots

When a polynomial has repeated or equal roots, it means that at least two of its roots are identical. This situation imposes strong conditions on the coefficients, often reflected by a zero discriminant or other specific algebraic relationships. Analyzing this case involves setting up equations from the fact that all roots are equal and utilizing Viète’s formulas to derive constraints on the coefficients.

Viète's Formulas

Viète's formulas connect the roots of a polynomial to its coefficients. In a cubic polynomial, these formulas express the sum of the roots, the sum of the products of the roots taken two at a time, and the product of the roots in terms of the coefficients. These relationships are fundamental in deriving conditions on the coefficients when the roots satisfy specific patterns like arithmetic or geometric progressions.

Arithmetic Progression

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant difference to the preceding term. When the roots of a polynomial are in arithmetic progression, their linear relation can be expressed in terms of a common difference. This leads to specific conditions among the sums and products of the roots, and consequently, to an equation relating the coefficients of the polynomial.

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