DIVISION BY DETACHED COEFFICIENTS.
63. In Art. 52 we considered certain cases of compound expressions in which the work of multiplication could be shortened by using the Method of Detached Coefficients. In the same cases the labor of division can be considerably abridged by using detached coefficients, and employing an arrangement of terms known as Horner's Method of Synthetic Division. The following examples illustrate the method :
Ex. 1. Divide 3x^5-8x^4-5x^3+26x^2-28x + 24 by x^3-2x^2-4x+ 8.
Divisor 1 3- 8- 5 + 26-28 + 24 Dividend 2 6 + 12-24 4 - 4 - 8 + 16 -8 + 6 + 12-24 Quotient 3 - 2 + 3 +0+ 0+0
Inserting the literal factors in the quotient according to the law of their formation, which is readily seen, we have for a complete quotient, 3 x^2 — 2 x + 3.
Explanation. The column of figures to the left of the vertical line consists of the coefficients of the divisor, the sign of each after the first being changed, which enables us to replace the process of subtraction by that of addition at each successive stage of the work. Dividing the first term of the dividend by the first term of the divisor, we obtain 3, the first term of the quotient. Multiplying 2, 4, and — 8, the remaining terms of the divisor, by this first term of the quotient gives the second horizontal line. We then add the terms in the second column to the right of the vertical line and obtain — 2, which is the coefficient of the second term of the quotient. With this coefficient as a multiplier, and using 2, 4, and — 8 again as a multiplicand, we form the third horizontal line. Adding the terms in the third column gives 3, which is the third term of the quotient. With this coefficient as a multiplier and the same multiplicand as before, we form the fourth horizontal line. As only zeros now appear in the quotient, the division is exact.
Ex. 2. Divide 2 a^ + 7 a^b + 12 a^b^ + 10 a^b^ -4 a^b^ by 2 a^ 4. 3 a^b — b^ to four terms in the quotient.
2 2 + 7 + 12 + 10 -4a^b^ + 0a^b^+ 0a^b^ -3 -3+ 0+ 1 0 -6+0 + 2 1 - 9 + 0+3 -3 +0 +1 Quotient 1 + 2 + 3 + 1 -5 + 3 + 1
