MULTIPLY every term of the multiplier by every term of the multiplicand, separately; setting down the products one after or under another, with their proper signs; and add the several lines of products all together for the whole product required. Note. In the multiplication of compound quantities, it is the best way to set them down in order, according to the powers and the letters of the alphabet. And in multiplying them, begin at the left hand side, and multiply from the left hand towards the right, in the manner that we write, which is contrary to the way of multiplying numbers. But in setting down the several products, as they arise, in the second and following lines, range them under the like terms in the lines above, when there are such like quantities; which is the easiest way for adding them up together. In many cases, the multiplication of compound quantities is only to be performed by setting them down one after another, each within or under a vinculum with a sign of multiplication between them. As (a+b) × (a−b) × 3ab, or a+b. a-b. 3ah. 8. Multiply 3a -2ax+5x2 by 3a2 —4ax-7x3. DIVISION. DIVISION in Algebra, like that in numbers, is the converse of multiplication; and it is performed like that of numbers also, by beginning at the left hand side, and dividing all the parts of the dividend by the divisor, when they can be so divided; or else by setting them down like a fraction, the dividend over the divisor, and then abbreviating the fraction as much as can be done. This will naturally divide into the following particular cases. CASE CASE I. When the Divisor and Dividend are both Simple Quantities; SET the terms both down as in division of numbers, either the divisor before the dividend, or below it, like the deno minator of a fraction. Then abbreviate these terms a much as can be done, by cancelling or striking out all the letters that are common to them both, and also dividing the one co-efficient by the other, or abbreviating them after the manner of a fraction, by dividing them by their common measure. Note. Like signs in the two factors make + in the quo tient; and unlike signs make-; the same as in multipli cation*, * Because the divisor multiplied by the quotient, must produce the dividend. Therefore, 1. When both the terms are +, the quotient must be +; because in the divisor in the quotient, produces + in the dividend. 2. When the terms are both, the quotient is also +; because in the divi sor in the quotient, produces in the dividend. 3. When one term is + and the other, the quotient must be; because ✦ in the divisor in the quotient produces—in the dividend, or in the divi bor, Xin the quotient gives-in the dividend. So that the rule is general; yiz, that like signs give +, and unlike signs give , in the quotient. CASE II. When the Dividend is a Compound Quantity, and the Divisor a Simple one: DIVIDE every term of the dividend by the divisor, as in the former case. 4. Divide 6ab-8ax+a by 2a. 5. Divide 3x-15+6x+6a by 3x. 6. Divide 6abc+12abx—9a3b by 3ab. 7. Divide 10a2x-15x2-25x by 5x. 8. Divide 15a2bc-15acx2+5ad2 by -5ac. 9. Divide 15a+3ay-18y2 by 21a. 10. Divide 20ab+60ab3 by --6ab, CASE III. When the Divisor and Dividend are both Compound Quantities; 1. SET them down as in common division of numbers, the divisor before the dividend, with a small curved line between them, and ranging the terms according to the powers of some one of the letters in both, the higher powers before the lower. 2. Divide the first term of the dividend by the first term of the divisor, as in the first case, and set the result in the quotient. 3. Multiply the whole divisor by the term thus found, and subtract the result from the dividend. 4. To this remainder bring down as many terms of the dividend as are requisite for the next operation, dividing as before; and so on to the end, as in common arithmetic. Note. |