Page:Elementary algebra (1896).djvu/483

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

$x^{3}-px^{2}+qx-r=0$.

find the value of

${\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}$.

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

$ry^{2}-qy^{2}+py-1=9$,

has for its roots

${\frac {1}{a}},{\frac {1}{b}},{\frac {1}{c}}$;

hence

$*\Sigma {\frac {1}{a}}$	$=$	${\frac {q}{r}},\Sigma {\frac {1}{ab}}={\frac {p}{r}}$.
⁂ $\Sigma {\frac {1}{a^{2}}}$	$=$	${\frac {q^{2}-2pr}{r^{2}}}$

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

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

a