and so on ; the divisor being x — y, the terms in the quotient all positive, and the index in the dividend either odd or even.
II. {x^ + y^} x + y = x^2 -xy +y^2, {x^+ y^} {x + y} = x^ — x^y + x^y^— xy^ + y^, {x^+y^ }{x + y } = x^- x^y + x^y^ - x^y^ + x^y ^- xy^+ y^ .
and so on ; the divisor being x + y, the terms in the quotient alternately positive and negative, and the index in the dividend alivays odd.
III. {x^-y ^} {x-y} =x + y , {x^ — y^} {x + y }=x^ — x^y + xy^ — y^, {x^ — x^y}{x + y }=x^ -x^y + x^y^ - x^y^ xy^ — y^.
and so on ; the divisor being x-y, the terms in the quotient alternately positive and negative, and the index in the dividend always even.
IV. The expressions x^-y^, x^+y^, x^+y^ \ldots. (where the index is even, and the terms both positive), are never divisible by x+y or x — y.
All these different cases may be more concisely stated as follows: (1) x^n -y ^n is divisible hy x -y if n be any whole number. (2) x ^n + y ^n is divisible by ;sx+ y if n be any odd whole number. (3) x^n -y^n is divisible by x+y if n be any whole number. (4) x^n + y^n is never divisible by x + y or x— y when n is an even whole number.
Note. General proofs of these statements will be found in Art. 106.
