Elements of Algebra

CHAP. III. OF ALGEBRA. 79 for if we subtract $6-2+4$ from $9-3+2$, we evidently obtain $9-3+2-6+2-4=0$; for $9-3+2=8$; also, $6-2+4=8$; and $8-8=0$. 267. Subtraction being therefore subject to no difficulty, we have only to remark, that if there are found in the remainder two or more terms, which are entirely similar with regard to the letters, that remainder may be reduced to an abridged form, by the same rules that we have given in addition. 268. Suppose we have to subtract $a-b$ from $a+b$; that is, to take the difference of two numbers from their sum: we shall then have $(a+b)-(a-b)$; but $a-a=0$, and $b+b=2b$; the remainder sought is therefore $2b$; that is to say, the double of the less of the two quantities. 269. The following examples will supply the place of further illustrations: $a^2+ab+b^2$ | $3a-4b+5c$ | $a^3+3a^2b+3ab^2+b^3$ | $\sqrt{a}+2\sqrt{b}$ $-a^2+ab+b^2$ | $2b+4c-6a$ | $a^3-3a^2b+3ab^2-b^3$ | $\sqrt{a}-3\sqrt{b}$ --- $2a^2$. | $9a-6b+c$. | $6a^2b+2b^3$. | $5\sqrt{b}$. CHAP. III. *Of the Multiplication of Compound Quantities.* 270. When it is only required to represent multiplication, we put each of the expressions, that are to be multiplied together, within two parentheses, and join them to each other, sometimes without any sign, and sometimes placing the sign $\times$ between them. Thus, for example, to represent the product of the two expressions $a-b+c$ and $d-e+f$, we write $(a-b+c) \times (d-e+f)$ or barely, $(a-b+c) (d-e+f)$ which method of expressing products is much used, because it immediately exhibits the factors of which they are composed. 271. But in order to shew how multiplication is actually performed, we may remark, in the first place, that to multiply a quantity, such as $a-b+c$, by $2$, for example,