Elements of Algebra

84 ELEMENTS SECT. II. It remains to multiply the product I. IV. by that of II. and III. $a^3+b^3$ $a^3-b^3$ --- $a^6+a^3b^3$ $-a^3b^3-b^6$ --- $a^6-b^6$ the same result as before. 281. It will be proper to illustrate this example by a numerical application. For this purpose, let us make $a=3$ and $b=2$, we shall then have $a+b=5$, and $a-b=1$; farther, $a^2=9$, $ab=6$, and $b^2=4$: therefore $a^2+ab+b^2=19$, and $a^2-ab+b^2=7$: so that the product required is that of $5 \times 19 \times 1 \times 7$, which is 665. Now, $a^6=729$, and $b^6=64$; consequently, the product required is $a^6-b^6=665$, as we have already seen. CHAP. IV. Of the Division of Compound Quantities. 282. When we wish simply to represent division, we make use of the usual mark of fractions; which is, to write the denominator under the numerator, separating them by a line; or to enclose each quantity between parentheses, placing two points between the divisor and dividend, and a line between them. Thus, if it were required, for example, to divide $a+b$ by $c+d$, we should represent the quotient thus; $\frac{a+b}{c+d}$, according to the former method; and thus, $(a+b) \div (c+d)$ according to the latter, where each expression is read $a+b$ divided by $c+d$. 283. When it is required to divide a compound quantity by a simple one, we divide each term separately, as in the following examples: $(6a-8b+4c) \div 2 = 3a-4b+2c$ $(a^2-2ab) \div a = a-2b$ $(a^3-2ab+3ab^2) \div a = a^2-2ab'+3b^2$