Elements of Algebra

86 ELEMENTS SECT. II. It is evident, in the first place, that the quotient will include the term $a$, since otherwise we should not obtain $a^2$. Now, from the multiplication of the divisor $a+b$ by $a$, arises $a^2+ab$; which quantity being subtracted from the dividend, leaves the remainder, $2ab+2b^2$; and this remainder must also be divided by $a+b$, where it is evident that the quotient of this division must contain the term $2b$. Now, $2b$, multiplied by $a+b$, produces $2ab+2b^2$; consequently, $a+2b$ is the quotient required; which multiplied by the divisor $a+b$, ought to produce the dividend $a^2+3ab+2b^2$. See the operation. $a+b)a^2+3ab+2b^2(a+2b$ $\underline{a^2+ab}$ $2ab+2b^2$ $\underline{2ab+2b^2}$ $0.$ 288. This operation will be considerably facilitated by choosing one of the terms of the divisor, which contains the highest power, to be written first; and then, in arranging the terms of the dividend, begin with the highest powers of that first term of the divisor, continuing it according to the powers of that letter. This term in the preceding example was $a$. The following examples will render the process more perspicuous. $a-b)a^3-3a^2b+3ab^2-b^3(a^2-2ab+b^2$ $\underline{a^3-a^2b}$ $-2a^2b+3ab^2$ $\underline{-2a^2b+2ab^2}$ $ab^2-b^3$ $\underline{ab^2-b^3}$ $0.$ $a+b)a^2-b^2(a-b$ $\underline{a^2+ab}$ $-ab-b^2$ $\underline{-ab-b^2}$ $0.$