ALGEBRA 



Form a third row by cross-multiplication 



n _ 2 01^2 #003 #1#4 #005 gjfle 000 7 



& 



#1 #1 0i 



Form a fourth row by operating on these last two rows by a similar cross- 

 multiplication. Continue this operation until there are no terms left. The 

 number of variations of sign in the first .column gives the number of roots 

 whose real parts are positive. 



If there are any equal roots some of the subsidiary functions will vanish. 

 In place of one which vanishes write the differential coefficient of the last one 

 which does not vanish and proceed in the same way. At the left of each row 

 is written the power of x corresponding to the first subsidiary function in that 

 row. This power diminishes by 2 for each succeeding coefficient in the row. 



Any row may be multiplied or divided by any positive quantity in order 

 to remove fractions. 



DETERMINATION OF THE ROOTS OF AN EQUATION 



1.260 Newton's Method. If a root of the equation f(x) = o is known to lie 

 between x\ and xz its value can be found to any desired degree of approximation 

 by Newton's method. This method can be applied to transcendental equations 

 as well as to algebraic equations. 



If b is an approximate value of a root, 



b JTTTT = c is a second approximation, 

 c j^-!- = d is a third approximation. 



This process may be repeated indefinitely. 



1.261 Horner's Method for approximating to the real roots of f(x) = o. 



Let pi be the first approximation, such that pi + i > c > pi, where c is the 

 root sought. The equation can always be transformed into one in which this 

 condition holds by multiplying or dividing the roots by some power of 10 

 by 1.231. Dimmish the roots by pi by 1.233. In the transformed equation 



A Q (X - pi) n + Ai(x - pl) n ~ l + ....+ An-l(* ~ pi} + A n = O 



put 



10 A n ,i 

 and diminish the roots by 2/10, yielding a second transformed equation 



H "~ 1 Bn = 



