12 Algebra. 



or a negative by a positive quantity, is 7iegative. It also follows 

 that in division like sig7is give plus and unlike sigfis give ?ni?ius. 

 The proof of these two statements is left as an exercise for the 

 student. 



26. Theorem. 77ie difference between like powers of two quanti- 

 ties is exactly divisible by the difference of the quafitities themselves. 



It is easily seen on trial that 



{a^ — jt:^)-i-(« — x) = a-^x. 

 (a^ — x^)-7-{a—x) = a''-\-ax-hx^. 

 (a* — x^)-^(a — x)=a^-{- a^'x-^ ax^ -{- x^. 

 (a^—x^)-7-(a—x)=a*-\-a^x+a^x^-\-ax^-{-x*. 

 Observing the uniform law in these results it would be at once 

 suggested that the theorem is universally true ; that is, that what- 

 ever be the value of n, 



^~'^=a"-'-\-a"-^x-^a"-'x'-^ . . +a'x"-'-j-ax''-'-j-x"-\ (i) 

 a—x 



This can easily be shown to be true, for multiplying the right 



hand side of this equation by a—x it becomes a"—x", as follows : 



a"-'-i-a"-'x-i-a"-'x'-\- . . . +a''x"-'-\- ax"-'+x"-' 



a — X 

 ~a'^~^a"^x+a"-'x'+ . . . -\-a^x"-'-\-a'x"-'-j-ax"-' 



—a"-'x—a"-'x^— . . . —a'x"-' —a'x"-'—ax"-'—x" 

 a" -}-o -f o -f- . ... +o +0 +0 —x" 



But multiplying the left hand side of equation (i) by a—x we 

 obtain a"—x" also. Hence equation (i) reduces to 



a" — x"=a" — x", 

 and hence must be correct. 



27. Theorem. The difference of like even powers is exactly 

 divisible by the sum of the quantities themselves. 



It will be found on actual division that 



{a" — x^)-^{a-\- x)=^a— X . 



{a*—x^)-i-(a-\-x)=a^—a^x-\-ax^—x^. 



(a'—x^)-7-(a-{-x)=a'—a*x+a'x'—a'x'-\-ax*—x\ 



