affected by the literal parts to which they are prefixed, these coefficients may be detached and written down with their signs in their proper order, and the multiplication performed as with polynomials. The partial products, numerical or literal, being carefully arranged as if undetached, are then reduced and the literal parts annexed. Now by annexing the proper literal parts to the several terms thus obtained, we have This method of multiplication may be employed when the two polynomials contain but one letter. 2. Multiply 3x-2x-1 by 3x+2. OPERATION. 3-2-1 9-6-3 +6-4-2 9+0-7-2 whence, or, 9x+0x2-7x-2, 9x-7x-2, Ans. When any of the powers of the letters, between the highest anu lowest, do not appear in either factor, the terms corresponding to such powers must be supplied, with the coefficient 0, 3. Multiply x+2x2-1 by x2+2. The factors completed are 3+2x2+0x-1 and x2+0x+2. Hence the operation is 1+2+0-1 4. Multiply 3x-2x-1 by 4x+2. Ans. 12x-2x2-8x-2. 5. Multiply 3x-5x-10 by 2x-4. 6. Multiply +xy+y' by x-xy+y'. Ans. 6x-22x2+40. 7. Multiply x3-4x2+5x-2 by x2+4x-3. Ans. x+xy2+yʻ. Ans. x-14x+30x3-23x+6. 441. Now, if detached coefficients can be used in multiplication. so in like cases, they may be employed for division. When the dividend and divisor contain but two letters and are homogeneous, the degree of the quotient will be the excess of the degree of the dividend over that of the divisor. EXAMPLES. 1. Divide a-3a'x-8a'x'+18ax +16x' by a-2ax-2x". 2. Divide a-5ab2+a2b3+6al'-26° by a3-3ab'+b3. In this example we must supply the term 0-ab in the dividend, and the term 0-ab in the divisor. The operation then is, 1+0-5+1+6-2 | 1+0-3+1 3. Divide x-4x-17x-13x2-11x-10 by x2+3x+2. When the dividend and divisor contain but a single letter, absent terms in either, answering to powers of this letter between the highest and lowest, must be inserted with the coefficient 0. In the examples we have wrought to illustrate the method of division by detached coefficients, the coefficients have been taken entire, that of the first term of the divisor, in each case, being unity; the process, however, will be the same whatever these coefficients may be. When the coefficient of the first term of the divisor is not unity, it may be made so by dividing both dividend and divisor by this coefficient. The quotient term will then be the first term of the corresponding dividend, as is seen in all the above examples. SYNTHETIC DIVISION. 442. To explain what synthetic division is, and to deduce a rule for executing it, let us take the first example in the preceding article. If the signs of the second and third terms of the divisor be changed, each remainder will be found, by adding the terms of the product of these two terms by the term of the quotient, to the corresponding terms of the dividend; observing that by the nature of the operation, the product of the first term of the divisor by the term of the quotient, cancels the first term of the dividend. Besides, since the first term of the divisor is unity, any quotient term is the same as the first term of the partial dividend to which it belongs. The process may now be indicated as follows: Hence the quotient is a-ax-8x2, as before found. The dividend and divisor are written in the usual way, after changing the signs of the last two terms of the latter; and a horizontal line is drawn far enough beneath the dividend for two intervening rows of figures. Bring down the first term of the dividend for the first term of the quotient. The products of the second and third terms of the divisor by the first term of the quotient are written, the first in the first row under the second term of the dividend, and the second in the second row under the third term of the dividend. The sum of the second vertical column is ther written for the second term of the quotient. The next step is multiply the second and third terms of the divisor by the second term of the quotient, placing the first product in the first row under the third term of the dividend, and the second in the second row under the fourth term of the dividend. The sum of the third vertical column is the third term of the quotient. The sums of the fourth and fifth columns each reduce to zero. The operation for the last example in the preceding article is 1—4—17—13—11-10|1-3-2 -3+21-6+15 -2+14-4+10 1-72-5 0 0 and for the quotient we have x3-7x+2x-5. No difficulty will now be experienced in understanding this general RULE.-I. If the coefficient of the first term of the arranged divisor is not unity, make it so by dividing both dividend and divisor by this coefficient. II. Write down the detached coefficients of the dividend and divisor in the usual way, changing the signs of all the terms of the of the latter except the first, and draw a line far enough below the dividend for as many intervening rows of figures as there are terms, less one, in the divisor, and bring down the first term of the dividend, regarded as forming a vertical column, for the first term of the quotient. III. Write the products of the second, third, etc., terms of the divisor by the first term of the quotient, beneath the second, third, etc., terms of the dividend in their order, and in the first, second, etc., rows of figures; and bring down the sum of the second vertical column for the second term of the quotient. IV. Multiply the terms of the divisor, exclusive of the first, as before, by the second term of the quotient, and write the products in their respective rows, beneath the terms of the dividend beginning at the third; bring down the sum of the third vertical column for the third term of the quotient. V. Continue this process until a vertical column is found of which the sum is zero, the sums of all the following also being zero when the division is exact; otherwise continue the operation until the desired degree of approximation is attained. Having thus found the coefficients of the quotient, annex to them the proper literal parts. In applying this method of division it is unnecessary to write the first term of the divisor, since it is unity and is not used in the oper ation. |