2. Multiply x-x2 by x3+x. 1+0-1 1+0+1 1+0-1 +1+0−1 1+0+0+0—1. With the letters and their powers added, it will be x2+0x+0x5+0x1—x3=x1—x3. The second, third, and fourth terms are of no value. 3. Multiply 3a-4ab2+6b3 by 2a2—4b2. 3+0-4+6 2+0-4 6+0-8+12 −12— 0+16—24 6+0-20+12+16-24. We now annex the letters with their proper powers, decreasing by a constant common difference, thus: 6a0ab-20a3b2+12a2b3+16ab1—24b3— 4. Multiply 2a3-3ab2+5b3 by 2a2—562. 2+0—3+ 5 2+0-5 4+0-6+10 -10+0+15-25 4+0-16+10+15-25. Affixing the letters with their powers, we have, 4a-16a3b2+10a2b3+15ab1-2563. 5. Multiply 5a-3a+a by 2a+a3. Ans. 10a-+5a3—6a®—a3+aa. 6. Multiply 3x3-2x-2 by x2-3. Ans. 3x-11x3-2x2+6x+6. 7. Multiply y2+y—3 by y3—y. Ans. y3+y'—4y3—y2+3y. 8. Multiply x+x1+x3+x2+x-+1 by x-1. 9. Multiply a2-2ab+462 by a2+2ab+462. Ans. x-1. Ans. a'+4a2b2+166*. 10. Multiply 3a+3a3b+3a2b2+3ab3+361 by 7a-7b. 11. Multiply 3+x2y+xy2+y3 by x-y. Ans. 21a-2165. Ans. x-y. SECTION V. DIVISION. ART. 88. Division is the converse of Multiplication, and is performed like that of numbers. Its object is to find how many times one quantity is contained in another; or to find what quantity, multiplied by a given quantity, will produce another given quantity. The product of like signs, as in the rule of Multiplication, produces, and unlike signs -. CASE I. 89. When the divisor and dividend are both simple quantities. If abc be divided by a, the quotient will be bc; because a multiplied by bc will produce abc. If 4abc be divided by 2a, the quotient will be 2bc; because 2a multiplied by 2bc will produce 4abc. If 9bx be divided by 3x, the quotient is 36; for 36 multiplied by 3x is '9bx. From the above illustration we derive the following RULE. Write 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 all the terms. Or, divide the coefficient of the dividend by the coefficient of the divisor, and cancel the letters common to the divisor and dividend. 90. Powers and roots of the same quantity are divided by subtracting the index of the divisor from that of the dividend. Thus, if we wish to divide a5 by a3, we subtract the index 3 from the index 5, and set the remainder 2 over the a; thus, a2. This process is evident from the fact that a3=aaaaa, and a3 =aaa, and aaaaa divided by aaa gives aa=a2. 91. When the divisor is a simple quantity, and the dividend a compound one, we adopt the following RULE. Divide each term of the dividend by the divisor, as in Art. 89. Or, we may write the divisor under the dividend, in the form of a fraction, and then cancel equal quantities when found in the divisor and in each term of the dividend. EXAMPLES. 1. Divide 9a3b+6u'c-12ab by 3a. OPERATION. 3a)9a3b+6a c-12ab 3a2b+2a3c-4b. Ans. We find that 3a is a factor in each term of the dividend; we therefore write the other factors under their respective quantities. 2. Divide 8abc+16a5bc-4a2c by 4a2c. 3. Divide 9abc-3a2b+18abc by 3ab. Ans. 2ab+4a3b—c. Ans. 3ac-a+6a2c. 4. Divide 20a bc+15abd3-10a be by 5ab. Ans. 4a3c+3d3—2ae. 5. Divide 15x3y3+30x3y' by x2. Ans. 6. Divide 7ax1yz3—14xyz+21xy2 by 7xy. Ans. 7. Divide p3mq+p3m—p1mc by p2. Ans. 8. Divide 4txz-8t2z+z2 by z. Ans. 9. Divide 12a-2-8a2b+16a3x-10a-2y by 2a2. Ans. 6a-4b+8ax-5a-y. CASE III. 92. When the divisor and dividend are both compound quan tities. RULE. Write down the quantities in the same manner as in the division of numbers in Arithmetic, arranging the terms of each quantity so that the highest powers of one of the letters may stand before the next lower. 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. 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 proceed, as in Arithmetic, till the work is finished. 3. Divide a1+4a2b2+16b1 by a2—2ab+4b2. |