18 BOOK OF MATHEMATICAL PROBLEMS. 



95. Prove that x 4 +px + q will be divisible by x 2 + ax + 6, if 



a 6 - 4<?a 2 = p*, and (b 3 + q) (b 3 - q) a =p s b*. 



96. The highest common divisor of p (x q 1) q (x p 1), and 

 (q-p) x q -qx q ~ v + p is (x- l) a , p, q being numbers whose greatest 

 common measure is 1, and q > p. 



97. If n be any positive whole number not divisible by 3, 

 the expression x 3n + I + (x + l) Sn will be divisible by x 2 + x + l' t 

 and, if n be of the form 3r + 2, by (x 3 + x + 1)*. 



Y. Identities and Equalities. 



98. Prove that, 



(1) (a + b + c) 3 = a 3 + I s + c 3 -f 3 (b + c) (c + a) (a + 5), 



2 J_ 2 (5-c) 8 + (c-^+(a-.5) 2 

 W 



(3) (6T-6 8 ) (S- c 2 ) + (^ 



= s(s-a)(s-b)(s-c), 

 where S/S' = a 2 + b* + c 2 , and 2s = a + b + c, 



(4) 2yV(* + x)'(x + y) 3 + 2 



x 4 (y + z) 4 + y 4 (z + x) 4 + z 4 (x + y) 4 + 1 Gofy V (yz + KX+XIJ), 

 (5) (6V + V 2 ) (5 - c) (a - d) + (c V + b*d 2 ) (c -a)(b- d) 



= (b-c)(c-a)(a-b)(d-a)(d-b)(d-c), 



(bed + cda + dab + abc) 2 - abed (a+b + c + d)' 

 = (be - ad) (ca - bd) (ab - cd), 



