Page:Elementary algebra (1896).djvu/483

Rh 586. To transform an equation into another whose roots are the reciprocals of the roots of the proposed equation.

Let $$f(x) = 0$$ be the proposed equation; put $$y = \frac{1}{x}$$, so that $$ x = \frac{1}{y}$$;  then the required equation is $$f\left(\frac{1}{y}\right) = 0$$.

One of the chief uses of this transformation is to obtain the values of expressions which involve symmetrical functions of negative powers of the roots.

Ex. If $$a, b, c$$ are the roots of the equation

find the value of

Write $$\frac{1}{y}$$ for $$x$$, multiply by $$y^2$$, and change all the signs; then the resulting equation

has for its roots

hence 587. Reciprocal Equations. If an equation is unaltered by changing x into {1}{x}, it is called a reciprocal equation.

If the given equation is

the equation obtained by writing {1}{x} for x, and clearing of fractions, is

If these two equations are the same, we must have

a
 * =z L stands for the sum of all the terms of which + is the type.