CASE III. When both the factors are compound quantities. RULE. Multiply every term of the multiplicand separately, by each term of the multiplier, setting down the products one after another, with their proper signs; then add the several lines of products together, and their sum will be the whole product required. 1. Required the product of x2-xy+y3 and x+y. 2. Required the product of x3+x2y+xy2+y3 and x-y. 3. Required the product of x2+xy+y2 and xa. xy+y2. 4. Required the product of 3x2-2xy+5 and 2+ 2xy - 3. 5. Required the product of 2a3-3ax+4x2 and 5a2 6αx-2x2. 6. Required the product of 5x3+4ax2+3a2x+a3, and 2x2-3ax+a2. 7. Required the product of 3x3+2x2 y2+3y3 and 2x3-3x2 y2+5y3. 8. Required the product of x3-ax2+bx-c and x2-dx+e. DIVISION. DIVISION is the converse of multiplication, 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 (e). It it 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. (e) According to the rule here given for the signs, it follows that as will readily appear by multiplying the quotient by the divisor; the signs of the products being then the same as would take place in the former rule. CASE I. When the divisor and dividend are both simple 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. 2a 3a Also-2a 3a, or =`~ &; and 9×3÷3x¥=3x\ 1. Divide 16x2 by 8x, and 12a2x2 by 2. Divide 2 8a2x. 15ay by 3ay, and 18ax2y by-8ax. 3 3. Divide—a bya, and axby-ax. 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. 1. Let 3x36x2+3ax-15x be divided by 3x. CASE III. When the divisor and dividend are both compound RULE. Set them down in the same manner as in division of numbers, ranging the terms of each of them so, that the higher powers 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. |