common to both of them, and supply any terms that may be wanting either in the divisor or dividend (to make the exponents of any common letter decrease or increase by the same constant difference as we pass from any term in either of them to the next successive term) by terms whose coefficients are 0, then if we omit the common letter (or letters) and use only the coefficients thus arranged, with the signs of the terms to which they belong in the division, we shall have what is called Division by detached coefficients. Having obtained the coefficients in the quotients, either by Rule I. or Rule II., we can easily supply the corresponding powers of the omitted letter or letters; for if we divide the power of any omitted letter in the first term of the arranged dividend by the power of the same omitted letter in the first term of the arranged divisor, we have the power of the letter in the first term of the quotient. The powers of any omitted letter in the following terms of the quotient may be supplied from the consideration that the powers of any omitted letter must follow the same law of increase or decrease in the terms of the quotient as in the terms of the arranged divisor and dividend. If we prefix the coefficients in the quotient with their proper signs to the powers of the omitted letter (or letters) that correspond to them, omitting those terms whose coefficients are 0, we shall have the quotient, as required. It is easy to see that the method of detached coefficients is nothing more than a simplification of Rules I. and II., the division being conducted in the same way, without putting down the powers of the common letter or letters according to which the divisor and dividend have been arranged, leaving the powers of the neglected letter or letters to be supplied in the quotient. (9.) There is a very simple method of Division by detached coefficients, which is called Synthetic Division, that appears to have been first discovered by W. G. Horner, Esq., of Bath, England. On account of its great importance in the solution of the higher equations, we shall now proceed to give it particular notice. HORNER'S SYNTHETIC DIVISION. 1. Having arranged the terms of the divisor and dividend as in the method of detached coefficients, we divide each coefficient in the divisor and dividend by the first coefficient in the divisor, and use the results as the detached coefficients in the divisor and dividend. 2. Then we divide the sum of the coefficients in the 'dividend by the sum of the coefficients in the divisor, as in Rule III., and we shall have the sum of the coefficients in the quotient; and by prefixing the coefficients in the quotient to the corresponding powers of the omitted letter or letters, we shall have the quotient of the given dividend and divisor, as required. We will now illustrate the method by a few examples. Ex. 1. To divide 3-81 by 3x-9. Here, by dividing the coefficients in the dividend and divisor by 3, the first coefficient in the divisor, they become 27 and 3; then, since and x, are wanting to the dividend, we supply the deficient terms by 0.2, 0.x, and the dividend becomes +0.2+0.27; and detaching the coefficients, we have 1+0+0-27 for the detached coefficients in the dividend, and the detached coefficients in the divisor are 1-3; and we have to divide 1+0+0,27 by 1-3, by Rule III. According to the rule, we must change the sign of the second term, -3, in the divisor, and we shall have 1+ 3 for the divisor. Hence, by Rule III., we must proceed as follows. 1+31+0+0-27 3+9+27 1+ 3+ 9 = the quotient, and the division is exact. Dividing the omitted power a3 of x in the first term of the given dividend by x in the first term of the given divisor, we have a for the power of a that corresponds to the first coefficient 1, in the quotient, and a evidently corresponds to 3, the second coefficient in the quotient; consequently, +3 +9 is the quotient arising from the division of 381 by 3x-9. Ex. 2.-To divide 6x6 12x+27x-10x2 + 20x — 45 by 4x + 9. Arranging the terms according to the descending powers of x, supplying the deficient term in the dividend, dividing the coefficients in the divisor and dividend by 2, the coefficient of the first term of the divisor, then detaching the resulting coefficients of the dividend and divisor, we shall have the sum of the coefficients 3 45 2 6 + 27 - + 0 5 + 10 2 9 2+ 2 to be divided by the sum of the coefficients 1 by Rule III.; and, according to the rule, changing the signs 9 of the last two terms of the divisor, we have 1 + 2 − g for the changed divisor. 2 Hence, according to the rule, we must proceed as follows. 3 + 0 + 0 + 0 5 + 0 + 0 = the coefficients in the quotient, and the division is exact. Dividing ", the power of x in the first term of the arranged dividend, by 2, the power of a in the first term of the arranged divisor, we have for the power of a that corresponds to the coefficient 3 in the first term of the quotient; and since the powers of a decrease by a unit as we pass from any term in the arranged divisor and dividend to the next successive term, it is clear that the sought quotient is expressed by 3x + 0.23 +0.x2+0.x-5=3x-5 by omitting the terms whose coefficients equal 0. Ex. 3.-Divide x3-3x1y2+3x2y-y by a3-3x2y + 3xy-y3. Arranging the dividend and divisor according to the descending powers of y, dividing the terms of the dividend. and divisor by 1, the coefficient of the first term of the divisor, supplying the terms that are wanting in the dividend, and detaching the coefficients, we have 1+0−3+0 +3+01 for the coefficients in the new dividend, and 1-3+3 -1 for those of the new divisor; and changing the signs of all the coefficients in the new divisor except that of the first term, we have 1 + 3 3 + 1 for the coefficients in the changed divisor. For convenience, we shall write the coefficients of the changed divisor in a vertical column, putting the first coefficient at the top, and the rest in their proper order. Then, by Rule III., we proceed as follows. 11+0-3+0+3+0−1 = the coefficients in the quotient; and supplying the powers of the letters that correspond to the coefficients, we evidently have y3 + 3y2x + 3yx2 + x3 for the quotient, as required. We will now perform the division by placing the products of the coefficients in the divisor and any term of the quotient, in diagonal lines, under the corresponding coefficients of the new dividend, which is in accordance with the directions given in Horner's rule. And since the first of the coefficients in the changed divisor is of no use in finding the coefficients in the quotient, we shall omit it. Hence, we shall have the following work. = the coefficients in the divisor; the same as found before. We think it will be evident from a comparison of this method of placing the products, with the method of placing them in horizontal lines (as we have done above), that it will generally be more simple to write them horizontally than diagonally, as directed in Horner's rule. - Ex. 4.-Divide adan+aex3n — bdx2 + aw3n-ben + c. We shall now suppose the terms of the dividend and divisor to be arranged according to the descending powers of ; then it is easy to see that the difference between the exponents of x in any two successive terms of the dividend is n; but the difference of the exponents of x in the first and second terms of the divisor is 2n; consequently there is a term wanting in the divisor between the first and second terms. It will also be noticed that the coefficients of the different powers of x in the dividend and divisor are letters; but it is easy to see that we may use them in the same way as if they were figures. Hence, if we divide the coefficients of the dividend and divisor by a, the coefficient of the first term in the divisor, supply the deficient term in the divisor, and detach the resulting coefficients; then, by Rule III., we must proceed as follows. = the coefficients in the quotient, and it is evident that the quotient is da" + e. It may be observed that the vertical bar is used in the given dividend to signify that cd-be is the coefficient of x"; also in the detached coefficients of the new dividend the vertical bar is used to signify that cd be a α is a coefficient in the new dividend. We have also written the detached coefficients of the changed divisor in a vertical column, for convenience and simplicity; and we have written the 0 in it on account of the deficient term in the divisor. Then, d is the first term of the quotient, which, multiplied b by 0, gives 0 to be placed under e, and, multiplied by α |