b3 5. Multiply a + ab + ab2 + 63 by a-b. - 462. Ans. 9a2 6. Multiply a + ab + b2 by a2 ab + b2. Ans. a-b1. 7. Multiply 3x2-2xy + 5 by x2 + 2xy — 6. 8. Multiply 3a-2ax+5x2 by 3a2-4ax-7x2. 9. Multiply 3x3 + 2x2y2 + 3y3 by 2x3-3x2y2 + 3y'. 10. Multiply a2 + ab + b2 by a-2b. 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 may naturally be distinguished into the following particular cases. 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 denominator of a fraction. Then abbreviate these terms as 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 quotient; and unlike signs make —; the same as in multiplication*. * Because the divisor multiplied by the quotient, must produce the dividend. Therefore, EXAMPLES. 1. To divide 6ab by 3a. Here 6ab3a, or 3a) 6ab ( or C 2. Also cc= = 1; and abx ÷ bxy = 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. 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 divisor X- 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 divisor X+in the quotient gives - in the dividend. So that the rule is general; viz. that like signs give +, and unlike signs give, in the quotient. When the Divisor and Dividend are both Compound 1. SET them down as in common division of numbers, the divisor before the dividend, with a small curved line between them, and range 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. If the divisor be not exactly contained in the dividend, the quantity which remains after the operation is finished may be placed over the divisor, like a vulgar frac tion, and set down at the end of the quotient as in arithmetic. 6. Divide 48z3. 96aza 64a3z+150a3 by 2z 3a. ALGEBRAIC FRACTIONS have the same names and rules of operation, as numeral fractions in common arithmetic; as appears in the following Rules and Cases. CASE I. To reduce a Mixed Quantity to an Improper Fraction. MULTIPLY the integer by the denominator of the fraction, and to the product add the numerator, or connect it with its proper sign, + or -; then the denominator being set under this sum, will give the improper fraction required. |