ALGEBRA 5 



the upper sign being used if the roots are to be diminished and the lower sign 

 if they are to be increased. The resulting equation will be: 



/v-2 /v-3 



f / 7 \ ft ( J \ i ft 9 / 7 \ f 1 1 1 / 7 \ 



where /'(*) is the first derivative of f(x), f"(x), the second derivative, etc. 

 The resulting equation may also be written : 



A Q x n + Aix n ~ l + A 2 x n ~ 2 + + A n -ix + A n = o 



where the coefficients may be found by the method of 1.222 if the roots are to 

 be diminished. To increase the roots by h change the sign of h. 



MULTIPLE ROOTS 



1.240 If c is a multiple root of f(x) = o, of order m, i.e.. repeated m times, 

 then 



/(*) = (x - cYQ; R = o 



c is also a multiple root of order m i of the first derived equation, f(x) = o; 

 of order m 2 of the second derived equation, f"(x) = o, and so on. 



1.241 The equation /(g) = o will have no multiple roots if f(x) and/'(#) have 

 no common divisor. If F(x) is the greatest common divisor of f(x) an.df(x), 

 f(x)/F(x) =fi(x), and/i(#) will have no multiple roots.' 



1.250 An equation of odd degree, n, has at least one real root whose sign is 

 opposite to that of a n . 



1.251 An equation of even degree, n, has one positive and one negative real 

 root if a n is negative. 



1.252 The equation f(x) = o has as many real roots between x = x\ and x = $2 

 as there are changes of sign in f(x) between xi and x*. 



1.253 Descartes' Rule of Signs: No equation can have more positive roots 

 than it has changes of sign from + to - and from - to +, in the terms of f(x). 

 No equation can have more negative roots than there are changes of sign in /(-*). 



1.254 If f(x) = o is put in the form 



A (X - ti) n + Ai(x ~ ti) n ~ l + + An = O 



by 1.222, and A Q , AI, , A n ^ are all positive, h is an upper limit of the 



positive roots. 



If f(-x) = o is put in a similar form, and the coefficients are all positive, 

 h is a lower limit of the negative roots. 



