Elements of Algebra

CHAP. IV. OF ALGEBRA. 85 ($4a^2 - 6a^2c + 8abc) \div 2a = 2a - 3ac + 4bc$ ($9a^2bc - 12ab^2c + 15abc^2) \div 3abc = 3a - 4b + 5c$. 284. If it should happen that a term of the dividend is not divisible by the divisor, the quotient is represented by a fraction, as in the division of $a + b$ by $a$, which gives $1 + \frac{b}{a}$. Likewise, $(a^2 - ab + b^2) \div a^2 = 1 - \frac{b}{a} + \frac{b^2}{a^2}$. In the same manner, if we divide $2a + b$ by $2$, we obtain $a + \frac{b}{2}$: and here it may be remarked, that we may write $\frac{1}{2}b$, instead of $\frac{b}{2}$, because $\frac{1}{2}$ times $b$ is equal to $\frac{b}{2}$; and, in the same manner, $\frac{b}{3}$ is the same as $\frac{1}{3}b$, and $\frac{2b}{3}$ the same as $\frac{2}{3}b$, &c. 285. But when the divisor is itself a compound quantity, division becomes more difficult. This frequently occurs where we least expect it; and when it cannot be performed, we must content ourselves with representing the quotient by a fraction, in the manner already described. At present, we will begin by considering some cases in which actual division takes place. 286. Suppose, for example, it were required to divide $ac - bc$ by $a - b$, the quotient must here be such as, when multiplied by the divisor $a - b$, will produce the dividend $ac - bc$. Now, it is evident, that this quotient must include $c$, since without it we could not obtain $ac$; in order therefore to try whether $c$ is the whole quotient, we have only to multiply it by the divisor, and see if that multiplication produces the whole dividend, or only a part of it. In the present case, if we multiply $a - b$ by $c$, we have $ac - bc$, which is exactly the dividend; so that $c$ is the whole quotient. It is no less evident, that $(a^2 + ab) \div (a + b) = a$; $(3a^2 - 2ab) \div (3a - 2b) = a$; $(6a^2 - 9ab) \div (2a - 3b) = 3a$, &c. 287. We cannot fail, in this way, to find a part of the quotient; if, therefore, what we have found, when multiplied by the divisor, does not exhaust the dividend, we have only to divide the remainder again by the divisor, in order to obtain a second part of the quotient; and to continue the same method, until we have found the whole. Let us, as an example, divide $a^2 + 3ab + 2b^2$ by $a + b$.