Elements of Algebra

CHAP. X. OF ALGEBRA. 111 $(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5$ $a+b$ $a^6+5a^5b+10a^4b^2+10a^3b^3+5a^2b^4+ab^5$ $a^5b+5a^4b^2+10a^3b^3+10a^2b^4+5ab^5+b^6$ $(a+b)^6=a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6, \ldots$ 342. The powers of the root $a-b$ are found in the same manner; and we shall immediately perceive that they do not differ from the preceding, excepting that the 2d, 4th, 6th, $\ldots$ terms are affected by the sign minus. $(a-b)^1=a-b$ $a-b$ $a^2-ab$ $-ab+b^2$ $(a-b)^2=a^2-2ab+b^2$ $a-b$ $a^3-2a^2b+ab^2$ $-a^2b+2ab^2-b^3$ $(a-b)^3=a^3-3a^2b+3ab^2-b^3$ $a-b$ $a^4-3a^3b+3a^2b^2-ab^3$ $-a^3b+3a^2b^2-3ab^3+b^4$ $(a-b)^4=a^4-4a^3b+6a^2b^2-4ab^3+b^4$ $a-b$ $a^5-4a^4b+6a^3b^2-4a^2b^3+ab^4$ $-a^4b+4a^3b^2-6a^2b^3+4ab^4-b^5$ $(a-b)^5=a^5-5a^4b+10a^3b^2-10a^2b^3+5ab^4-b^5$ $a-b$ $a^6-5a^5b+10a^4b^2-10a^3b^3+5a^2b^4-ab^5$ $-a^5b+5a^4b^2-10a^3b^3+10a^2b^4-5ab^5+b^6$ $(a-b)^6=a^6-6a^5b+15a^4b^2-20a^3b^3+15a^2b^4-6ab^5+b^6, \ldots$ Here we see that all the odd powers of $b$ have the sign $-$, while the even powers retain the sign $+$. The reason