2 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



21. (a -; h + c]( - c 4' b -f c)(a -b + c}(a + b - c) = 2(b 2 c 2 + cV + a?b 2 ) 



-(a 4 + 6 4 + c 4 ). 



22. (a + 6 + c + d) 2 + (a + b - c - d) 2 + (a + c - b - d) 2 + (a + d - b - c) 2 



= 4 (a 2 + b 2 + c 2 + d 2 ). 

 If A = aa + by + c/3 

 .B = a/3 + a + 7 

 C = ay + b@ + ca 



23. (a + 6 + c)(a + ^ + 7)=^+5 + C. 



24. [a 2 + 6 2 + c 2 - (a& + 6c + cfl)] [a 2 + p + y* - (aft + ^7 + 7 a)] 



= A 2 + 5 2 + C 2 - (45 + C + C4). 



25. (a 3 + b 3 + (? - 3abc)(a* + ft 3 + 7 3 - 301/37) = 4 3 + 5 s + C 3 - 



ALGEBRAIC EQUATIONS 



1.200 The expression 



f(x) = aox n + a^- 1 + a 2 ^ n ~ 2 + ...... + a n ^x + a n 



is an integral rational function, or a polynomial, of the nth degree in x. 



1.201 The equation J(x) = o has n roots which may be real or complex, dis- 

 tinct or repeated. 



1.202 If the roots of the equation f(x) = o are c\, c 2 , . . ., c, 



f(x) = ao(x - ci)(x - cz) ...... (x - c n ) 



1.203 Symmetric functions of the roots are expressions giving certain com- 

 binations of the roots in terms of the coefficients. Among the more important 

 are: 



Ci + C 2 + .......... + C n = -- 



a 



CiC 2 + CiCz + . . . + C 2 C 3 + CzC* + ...... + Cn-lCn = ~ 



a 





C n = (-l) n 



7 1.204 Newton's Theorem. If s k denotes the sum of the &th powers of all the 

 roots of f(x) = o, 



k k k 



101 + SidQ = O 



2dz + S&i + S 2 dQ = O 



3^3 + ^1^2 + s z ai + 



