The number of real roots of the equation x^3-(1)/(x^3)+4(x-(1)/(x))=0, x ≠0 (a) 3 (b) 0 (c) 1 (d

step 1 So, in the given question we are told that we have an equation of the form x cube, x cube minus

The number of real roots of the equation...

Key Concepts

Change of Variable Technique

This method involves substituting part of the expression, for example setting u = x - 1/x, to take advantage of patterns and simplify equations that combine functions of x and its reciprocal. This transformation is especially useful for reducing the complexity of the original equation, converting a rational or mixed function into a polynomial form that is easier to analyze.

Symmetric Functions Involving Reciprocals

Equations that include terms like x and 1/x exhibit symmetry because replacing x with 1/x leaves the structure of the expression largely unchanged. Recognizing this symmetry can lead to strategic substitutions that exploit the inherent balance in the equation, thereby simplifying it and revealing underlying algebraic identities.

Polynomial Reduction and Factoring

By using appropriate substitutions, the original equation can be transformed into a polynomial equation. This reduction process makes it possible to apply standard methods of factoring and polynomial analysis, which simplify the task of determining the number and nature of the roots.

Real Root Analysis in Transformed Equations

Once the equation has been reduced to a polynomial form, techniques from real analysis can be employed to assess the number of real solutions. This involves studying the behavior of the polynomial function, such as its turning points and sign changes, to determine how many roots lie on the real number line.

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