9. Required the product of the four following factors, viz. I. II. III. IV. (a + b), (a2 + ab + b2), (a - b), and (a2 — ab + b2). Ans. a6 66. ენ. 10. Required the product of a3+3a2x + 3ax2 + x3 and a3-3a2x+3ax2 x3 Ans. a6 3a2x2+3a2x2 42 11. Required the product of a + a2c2 + c2 and a2 - c2. Ans. a 6 C6 12. Required the product of a + b2 + c2 — ab and a+b+c. Ans. a3 3abcb3+ c3. ac - be DIVISION. DIVISION IS the converse of mulplication, and is performed like that of numbers; the rule being usually divided into three cases; in each of which like signs give + in the quotient, and unlike signs, as in finding their products.† +ab b It is here also to be observed, that powers and roots of the same quantity, are divided by subtracting the index of the divisor from that of the dividend. I a2 2 aðað, or ==— a3; and a + a", or "= a 4 1 12 ат J , aman, 2 a3 an L a3 am-n CASE I. When the divisor and dividend are both simple quantities. RULE.-Set the dividend over the divisor, in the manner of a fraction, and reduce it to its simplest form, by cancelling the letters and figures that are common to each term. * I would advise the learner to perform the calculation of this example several ways; viz. First, by multiplying the product of the factors I. and II. by the product of the factors III. and IV. Secondly, by multiplying the product of the factors I. and III. by the product of the factors II. and IV. Thirdly, by multiplying the product of the factors I. and IV. by the product of the factors II. and III. The last method is the most concise. See Euler's Algebra, page 119, Vol. I.--ED. † According to the rule here given for signs, it follows that - ab +ab 09 - ab --b =+a, α, +a, b +b as will readily appear by multiplying the quotient by the divisor; the signs of the product being then the same as would take place in the former rule. 6ab2a, or ɑ — ɑ, or a a 6ab 2a 3. Divide 1; and a÷ ૭૨ ૭ 3b; and 12ax2 - 3x, or -2a 2 1 EXAMPLES. a 2 2a÷÷÷ 3a, or 3; = 4 8a2x Ans. 2x, and 8ax. 5y, and 1 1 1 2 A 3, and ax by — 23aa. 3a 1. Divide 16x2 by 8x, and 12a2x2 by 2. Divide 15ay2 by 3ay and 15ay by 3ay and (ab + b2)÷÷ 2b, or ab + b2 26 (10ab15ax) ÷ 5a, or — EXAMPLES. a 10ab Ans. 31, and 4. Divide 12ab2 by — 3ab, and - 15ay by - 3ay2. 용 1 1 2 Ans. - 46, and 5y. 5. Divide 15a2x2 by 5ax2, and 21a2c2x by — 7ac2x4. 3a, and 3αx 4. 17x31a3c by — 5x31a2c1, and 24xy by 8 ✓ (xy), Ans. 6. Divide — 17% ac by 1 1. 17x6 ac2 Ans. 5 =a+b= 15ax 1. , 12ax2 3x CASE II. When the divisor is a simple quantity, and the dividend a compound one. RULE.-Divide each term of the dividend by the divisor, as in the former case; setting down such as will not divide in the simplest form they will admit of. 26 3x. 4ax, and 3 √(xy) (30ax 48x2)6x, or 1. Let 3x2+ 6x2 + 3ax 2. Let 3abc + 12abx 3. Let 40a3b3 + 60a2b2 17ab be divided by 12x2 30ax 48x2 5α 6x 15x be divided by 3x. 9a2b be divided by 3ab. 8x Ans. x2+2x + α -5. 2 xy + y2 2 Ans. 3ab + 5 5. Let 20ax2 + 15ax2 + 10ax +5a be divided by 5a. Ans. 4x3-3x2 + 2x + 1. 6. Let 6bcdz + 4bzda — 2b31⁄22 be divided by 2bz. Ans. 3cd+ 2ď2 bz. 7. Let 14a2 EXAMPLES. x + y) x2 + 2xy + y2(x + y x2 + xy 2 12a2b2 be divided by Ans. 2a b + 3x - 4 d2 4ab. Ans. 5- 1562 + 3ab. CASE III. When the divisor and dividend are both compound qualities. RULE.-Set them down in the same manner as in division of numbers, ranging the terms of each of them so, that the higher power of one of the letters may stand before the lower. Then divide the first term of the dividend by the first term of the divisor, and set the result in the quotient, with its proper sign, or simply by itself, if it be affirmative. This being done, multiply the whole divisor by the term thus found; and, having subtracted the result from the dividend, bring down as many terms to the remainder as are requisite for the next operation, which perform as before; and so on, till the work is finished, as in common arithmetic. a + x) a3 + 5a2x+5ax2+x3 (a2 + 4xx +x2 a3 + a2 x 2 ax2 + x3 9x-27 9x 27 9a2x2 + 6a3x a + x) a3 — x3 (a2 — ax + x2 203 2 a3 + a3 x ეიპ 2 ax 2 2 11a2x2+6a3x 2 2 J 6x+9 a2 (2x2 + 3ax NOTE 1. If the divisor be not exactly contained in the dividend, the quantity that remains after the division is finished, must be placed over the divisor, at the end of the quotient, in the form of a fraction; thus,* 2003 2x3 * In the case here given, the operation of division may be considered 4 x + y) x2 + y1 (x3 — x2y + xy2 · — y3 + . x2 + x3y 4 x3y + y1 x2 y2 + y2 Qy1 2. The division of quantities may also be sometimes carried on, ad infinitum, like a decimal fraction; in which case a few of the leading terms of the quotient will generally be sufficient to indicate the rest, without its being necessary to continue the operation; thus, a.+x) a. (1 the third term, &c. a ཆ། a + a2 3* as terminated, when the highest power of the letter, in the first or leading term of the remainder, by which the process is regulated, is less than the power of the first term of the divisor; or when the first term of the divisor is not contained in the first term of the remainder; as the succeeding part of the quotient, after this, instead of being integral, as it ought to be, would necessarily become fractional. x5 x6 * Now, it is easy to perceive that the next or 6th term of the quotient will be and the seventh term and so on, alternately plus and a5' αδ minus; this is called the law of continuation of the series. And the sum of all the terms when infinitely continued is said to be equal to the 2 a • a + x fraction Thus we say the vulgar fraction when reduced to 9' a decimal, is 22222, &c., infinitely continued. The terms in the quotient are found by dividing the remainder by a, the first term of the divisor; thus, the first remainder x divided by a, gives the second C a &c.* x2 term in the quotient; and the second remainder + gives a x2 + a2 |