EXAMPLES. 1. Divide 1- by 1+x. Ans. 1-2x+2x–2x+etc. 2. Divide 1 by 1+.. Ans. 1-x+x-+x-etc. 3. Divide ao_5a*x+10aox_10a*x* +5ax*—xby a'—2ax+x'. Ans. a'430'x+3ax'—'. 4. Divide x–5x+15x*—24x*+27x*—13x+5 by x*—2x9+ 4x*_-2x+1. Ans. x*—3x+5. 5. Divide r'-y' by x-y. Ans. x+a'y+x*y*+#oy' +x*y* +xyo+yo. 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. 1. Transform the equation x_4x*—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: 1-4 +0 8+322 8-32 X'. 2 2 2 4 2.3 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 xt12.c+17.-9.+7 = 0, into another whose root shall be 3 less. 0 = 3 X'. 2.3 Hence the transformed equation is 4+0y-37y - 123y-110 =0. Transform the equation #—12x—28 -- 0, into another whose roots shall be 4 less. Make x=y+4. OPERATION. 10 -12 -28 (4 4 +16 +16 X 2 Hence the transformed equation must be yo+12y'+367—12=0. Transform the equation x-10x* +30-6946 = 0, into another whose roots shall be less by 20. We make x = 20+y. 2 12 = The three remainders are the numbers just above the double lines, which give the following transformed equation : y' +50yo +8034—2886 = 0. Transform this equation into another whose roots shall be less by 3 Put y = 3+. 3 159 +2886 962 0 3 168 56 1130 59 Hence the transformed equation is 2+59z+1130% = 0. This equation may be verified by making z = 0; which gives y = 3, and x = 20+3=23. 444. If the signs of the alternate terms of any complete equation involving but one unknown quantity be changed, the signs of all the roots will be changed. In the general equation *" +A+B.c +...+Tx+U=0, (15 let the signs follow each other in any order whatever. Changing the signs of the alternate terms of this equation, beginning with the second, we have 2.-A...+B.cm-- .... +Tc+U=0; (2) hut if the change begin with the first term, we have -2cm + Axm-1-B.xm-2+. I Tx+U=0. (3) Now, if a be a root of equation (1), its first member reduces to zero when a is substituted for x; that is, the sum of the positive terms becomes equal to the sum of the negative terms. But if -a be substituted for x in equations (2) and (3), the numerical values of the terms of these equations will be equal to the values of the corresponding terms of equation (1), while the signs of the terms in equation (2), if m is an even number, will be the same, and those of equation (3), opposite to the signs of the terms of like degree in equation (1). If m is an odd number, the reverse will be true in respect to signs. In either case however, if a is a root of equation (1), -a is a root of both equation (2) and equation (3). An obvious consequence of this proposition is, that the roots of an equation are not affected by changing the signs of all its terms. EXAMPLES. 1. The roots of the equation --7x* +13:43= 0, are 3,2+3, and 2-V3; what will be the roots of the equation x + 7xo+ 13.x +3=0? Ans. -3, -2-V3, -2 +V3. 2. The roots of the equation X* — 3x® + 3x° +17.x— 18==0, are 1, 2, 2+1=5, and 2-V=5; what are the roots of the equation x+3x+3.r'-17.–18 = 0? Ans. -1, +2, -2–V-5, -2+135. 445. If all the coefficients of an equation be real and rational, surd and imaginary roots can enter the equation only by pairs. Let the coefficients A, B,.... U, of the equation 30* + A1+B.cm-+....Tc+U=0, (1) be all real and rational, and suppose that a+Vb is one of the roots of this equation. Substituting this value for x, we have (a+V+6)”+A(a+V+6)-1Ba+V =3)–2+.... = 0. (2) +T(a+V+6)+ U Expanding the several terms of this equation by the binomial formula, we have Ta+TV+, U; observing, in reference to the final terms, +(VEB)", +(V+)--, etc., that the sign + is to be used before those only which have odd numbers for their exponents; when the exponent is even, the plus sign is to be understood. If the root of equation (1) be atıb, the aggregate of these developments will be composed of two parts, the one rational and the other surd. The rational part will be the algebraic sum of those terms which have the even powers of vb for factors, the zero pow. er being included. Represent this part by M. |