Resolution of Algebraical Equations. 131 



When the given equation contains fractional or negative powers 

 of x, put (as was before observed) x = z + a, and considering as as 

 the unknown quantity, we may directly apply the method. 



The rule given in this section may be thus proved. Suppose 

 <p (x) to be resolved into its simple factors; i.e. 



<f>(x) = C .(x-a) .(x- )3) .{X - y) 



where C" = C.(-/3) . (- 7 ) . & c . 

 $(x) 



L«H-ie t l(l-3 + L(l-S) + L(i-f) t 



&c. 



(iJ ' -\ y) 



of which the only term which contains negative powers of x, is 

 the coefficient of - therefore in 1. ^-^ is - a, where a is a root of 



X X 



the equation (p (x) = o. 



Supposing the equation of n dimensions, which of its roots is 

 given by this method ? The result analytically comprehends all 

 the roots, but arithmetically it gives only the least root. Let us 

 recall, for example, the quadratic equation x" + ax + b = 0; and, 

 supposing a, /3, to be the roots; -« = a + /3, i = «/3; and substituting 

 these values in the expression for the root given at the beginning 

 of this Section, we get 



a/3 a c /3 2 4 a 3 /3 3 6.5 a* & 



root = 7^ + faTW + 2-faTW s + ¥r 3 -(^TW + Slc - 



r2 



