Elements of Algebra

102 ELEMENTS SECT. II. 323. When there is a remainder after the division, it is a proof that the root is composed of more than two terms. We must in that case consider the two terms already found as forming the first part, and endeavour to derive the other from the remainder, in the same manner as we found the second term of the root from the first. The following examples will render this operation more clear. $a^2+2ab-2ac-2bc+b^2+c^2 (a+b-c$ $a^2$ --- $2a+b) 2ab-2ac-2bc+b^2+c^2$ $2ab \qquad +b^2$ --- $2a+2b-c) -2ac-2bc+c^2$ $-2ac-2bc+c^2$ --- $0.$ $a^4+2a^3+3a^2+2a+1 (a^2+a+1$ $a^4$ --- $2a^2+a) 2a^3+3a^2$ $2a^3+a^2$ --- $2a^2+2a+1) 2a^2+2a+1$ $2a^2+2a+1$ --- $0.$ $a^4-4a^3b+8ab^3+4b^4 (a^2-2ab-2b^2$ $a^4$ --- $2a^2-2ab) -4a^3b+8ab^3+4b^4$ $-4a^3b+4a^2b^2$ --- $2a^2-4ab-2b^2) -4a^2b^2+8ab^3+4b^4$ $-4a^2b^2+8ab^3+4b^4$ --- $0.$