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: 1-3-8+18+16 1+2+2 Hence the quotient is a2-ax-8x', 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 x-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. Z 1. Divide 1-x by 1+x. 3. Divide 1 by 1+x. EXAMPLES. Ans. 1-2x+2x-2x+etc. Ans. 1-x+x-x+x*—etc. 3. Divide a'-5a‘x+10a3x2-10a3x3+5ax*—x* by a3—2ax+x3. Ans. a-3a'x+3ax'—x3. 4. Divide x-5x+15x-24x+27x-13x+5 by x-2x+ 4x3-2x+1. Ans. x3-3x+5. 5. Divide x'y' by x—y. Ans. x+xy+x*y*+x*y*+x*y*+xy°+y°. 443. The transformation of an equation into another having roots less or greater than those of the given equation by a fixed quantity, may now be expeditiously made by the method of synthetic division. i. Transform the equation x-4x3—8x+32 = 0, into another whose roots shall be less by two. The second power of x not appearing in this equation, it must be introduced with 0 for its coefficient. Instead of keeping the above operations separated, they may be united and arranged as follows: To understand this, it is only to be borne in mind that the divisor is the same throughout, and that the first term, 1, of the successive dividends, which if written would all fall in the vertical column at the left, is omitted. Transform the equation x-12x+17x2—9x+7 = 0, into another whose root shall be 3 less. Hence the transformed equation is y'+0y'-37y-123y-1100. Transform the equation x3-12x-280, into another whose roots shall be + less. Make x = = y +4. |