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Reciprocal Roots

To transform an equation into another whose roots are the reciprocals of the roots of the given equation.

Let

and hence the transformed equation is

Thus the equation whose roots are reciprocals of the roots of the given equation is obtained by putting

The chief use of this transformation is to obtain the values of the expression which involve symmetric functions of the negative powers of the roots.

Let

**ƒ(x) = 0**, then the equation of transformation isand hence the transformed equation is

Thus the equation whose roots are reciprocals of the roots of the given equation is obtained by putting

**1/x**for**x**in the given equation.**Reciprocal equation:**The equation**ƒ(x) = 0**is said to be reciprocal if it remains unaltered when**x**is replaced by**1/x**, i.e. in other words, the equation**ƒ(x) = 0**is a reciprocal equation if**For example: x**, is a reciprocal equation.^{4}– 4x^{3}+ 8x^{2}– 4x + 1 = 0The chief use of this transformation is to obtain the values of the expression which involve symmetric functions of the negative powers of the roots.

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