or, multiplying by x", dividing by U, and reversing the order of the terms, 234. 3d. To multiply the roots of a given equation by any quantity whatever. Let the quantity be n, and the given equation T + -+U=0; the problem requires that the roots of the transformed equation shall be the products which are obtained by multiplying all the roots of equation A by n. Therefore, y being the new unknown quantity, y=nx or x= Consequently equation A is changed into It is evident that the substitution of na for y in this equation gives the same result as the substitution of a for x in equation A; so that a cannot be a root of the equation in x without the product na being a root of the equation in y. Multiplying all the terms of the transformed equation by n", y+Pny+Qn2yTM—2, .... +Tn"-y+Un"=0 (B). If the coefficients of the proposed and transformed equations are compared, it becomes evident that the latter are formed by multiplying the former by no, n1, n2, &c. respectively, so that in each term the sum of the exponents of n and y is equal to m. This transformation comprehends that in which it is proposed to divide the roots by a number, n; for this amounts to the multiplication of the roots 1 n by and by consequence to the division of the coefficients of the given equation by no, n1, n2, &c. 1st example: Multiply the roots of the equation *—12x3+17x2—9x+7=0 by 2. In this example no=1, n1=2, n2=4, n3=8, n1=16; and the transformed equation is xa—24x3+68x2—72x+112=0. 2d example: Divide the roots of the equation r3+3x3-12x+432 by 6? n°=1, n1=}, n2=35, n3=zio ; and the transformed equation is x3+x-x+2=0. 235. 4th. To transform an equation which has fractional coefficients into another of which the coefficients shall be whole numbers, and that of the first term 1. Leaving the quantity n indeterminate, let nx=y, and let the transformation of the last Article be effected. The coefficients P, Q, &c. being fractional, let a common multiple, M, of the denominators be formed; then in equation B take n= -M; the coefficients Pn, Qu2, &c. will thus be rendered integer. In practice it is sufficient to take the least common multiple of the denominators. 3x2 5x 2 Example: It is required to transform the equation a3-2+49=0 into an equation whose coefficients are integer, and that of the first term 1. The least common multiple of the denominators is 4×9=36. Making n=36 in formula B, 3 × 36 × x2 5×362 × x 2×363 2 + 4 =0; 9 or x3-54x2+1620x-10368=0. 236. 5th. To augment or diminish the roots of a given equation by a given quantity. Let the given quantity be a, and the given equation x+Px-1+Qx-2 +Tx+U=0 (A). Making xy+a, y=x-a; consequently, by substituting y+a for x in equation A, a transformed equation is obtained, of which the roots are those of equation A diminished by a. In order that the roots of equation A may be augmented by the quantity a, it is necessary to change a into -a. Effecting the substitution of y+a for x, equation A becomes (y+a)+P(y+a)m−1+Q(y+a)TM-2 .. +T(y+a)+U=0. Developing the expressions (y+a)TM, (y+a)TM−1, &c. m(m-1) y”+ma y”-1+ 2 If it is required that the exponents of y shall increase from left to right, it is necessary to substitute a+y instead of y+a in equation A. The result is A'=ma-1+(m—1)PaTM~2+(m—2)Qa”—3 A"=m(m-1) a−2+(m−1)(m—2)PaTM−3+(m−2)(m—3) QaTM—4 A""=m(m—1)(m—2)aTM−3+(m−1)(m—2)(m—3)PaTM→+ .... &c. &c. .... The quantities A, A', A". are connected with the proposed equation X=0 by a very simple law. A is formed by writing a instead of x, in the first member X. A' is formed from A by multiplying each term of A by the exponent of a in that term, and diminishing the exponent of a by 1. As the last term is equivalent to Uao, it must vanish in the operation, being multiplied by zero. A' is also A" is formed in the same manner from A', A"" from A", &c. The quantities A', A”, A'" . .... are termed Derived Polynomials. A' is said to be the derived polynomial of A, A" of A'... designated the first derived polynomial of A, A" the second, &c. Example: Let it be required to substitute a+y for x in the equation x3-2x2-4x+5=0. A=a3-2a-4a+5. A'=3a2-4a-4. A"=6a-4. A"=6. Consequently, taking care to divide A" by 2, and A"" by 2×3, the transformed equation is (a3-2a2-4a+5)+(3a2-4a-4)y+(3a−2)y2+y3=0. If a is a root of the equation X=0, this equation is divisible by x-a, and the depressed equation is of the form The coefficients P', Q', &c. can be formed from the coefficients P, Q, &c. of the proposed equation, and from each other, by a process explained in Article 231. If a is not a root, the proposed equation is not exactly divisible by x-a. Effecting the division as far as possible, the result, which is of the same degree (m) as the proposed equation, is of the form M-1 +A'x'+A=0, in which r'=x-a; that is, in which the roots are those of the proposed equation diminished by a, and the coefficients are the Mth, M-1th, . second, first derived polynomials of A. If the proposed equation is numerical, the coefficients of the equation in x can be obtained from the formulæ A=a+Pa", &c.; A'=ma"-1+(m−1)Pa"-2, &c. &c. For example, let it be required to transform the equation x2+5x+4x2+3x-105=0 into another equation whose roots are less by 2 than the roots of the proposed equation. The transformed equation is of the form A" A" -x^2+A'x'+A=0. 2 A=24+5×23+4×22+3×2-105-16+40+16+6-105=-27. 48+60+8 116 = 6 =13. 2x3x4×2+2×3x5 48+30 78 2×3 Therefore the transformed equation is x+13x+58x2+111x-27=0. The following rule, founded on observation of the law according to which the coefficients of the transformed equation are composed of the coefficients of the proposed equation and the quantity a which expresses the difference between and y, is given by Mr. Horner in his Method for the General Resolution of Numerical Equations: "1st. To the first coefficient add the correction 0, and to each succeeding coefficient add a times the amount of the preceding coefficient and its correction; proceed thus with all the coefficients in order till the absolute term is corrected. "2d. Apply a similar process to the series of corrections, and then to those which are deduced from them; and so on, as often as it can be done, stopping each process one term short of the preceding. "3d. Add up each column of additional corrections, together with the once-corrected coefficient under which it stands; and the sum will be the new coefficients, standing vertically under those given ones for which they are to be substituted." The transformation of the last example is effected by this rule as follows: 1, + 5, +4, +3, -105 78 The last results taken in order are the coefficients of the transformed equation, which, as already found, is, x+13x3+58x2+111x-27=0. Explanation. In this example, a=2. The first line of the calculation contains the coefficients of the proposed equation and the term wholly known. The second and third lines are formed in this manner. O is the correction of the first coefficient, 1+0=1. This 1 is the first term of the third line. This result 1, multiplied by 2=2. This 2 is written under the second coefficient, 5, to which it is added. The sum +7 is written in the third line, of which it is the second term. This 7 is multiplied by 2; the product 14 is written under the third coefficient, +4, to which it is added. The sum +18 is the third term of the third line. This 18 is multiplied by 2; the product 36 is written under the fourth coefficient, +3, to which it is added. The sum 39 is the fourth term of the third line. This 39 is multiplied by 2; and the product 78 is written under the absolute term. The sum -27 is the fifth and last term of the third line. The fourth line is formed from the third and second thus ; 1, the first term of the third line, multiplied by 2=2. This 2 is written under the second term of the third line. This 2 multiplied by 2=4; the product 4 is added to 14, the third term of the second line; and the sum 18 is written under the third term of the third line. This 18 is multiplied by 2; and the product 36 added to 36, the fourth term of the second line. The sum 72 is written under the fourth term of the third line. The fifth line is formed from the fourth; thus, 1 (which in this example may be considered as the first term of each line) multiplied by 2=2. The 2 is written under the second term of the fourth line. This 2 is multiplied by 2; and the product 4, being augmented by 18, the third term of the fourth line, the sum 22 is written under the third term of the fourth line. The sixth line is formed from the fifth, as the fifth from the fourth, and so on. The sum of the third, fourth, fifth, sixth . . . lines is contained in the last line. The numbers in the last line are the coefficients of the transformed equation. 2d Example. It is required to transform the equation, x5-8x++11x3-39x2+31x-101=0, into an equation whose roots shall be less by 4 than the roots of the proposed equation? The transformed equation is x5+12x4+43x′3-35x^2-521-921. 3d Example. It is required to transform the equation, 2x-13x2+10x-19=0, into an equation whose roots shall be less by 3 than the roots of the proposed equation? G G 450 In this example the coefficient of the term in 23 is 0, and a=3. 2, + 0, -13, +10, -19 0, + 6, +18, +15, +75 The transformed equation is, 2x++24x3+95x^2+148x′+56=0. 4th Example. It is required to transform the equation, 6x3-3x2+4x-1=0, into another equation, whose roots shall be greater by 3 than the roots of the proposed equation? In this example, a=—3. 3, +4, 6, - 0, -18, 63, -201 6, 21, 67, -202 -18, +117 -18 6, 57, +184, -202 The transformed equation is, 6x3-57x2+184x′—202=0. By making the additions as well as the multiplications mentally, and recording only the results, the calculation of the coefficients of a transformed equation, by this method, may be abridged in appearance. The process for the second example (which is the longest) may be abridged as follows: The first line as before contains the coefficients of the proposed equation; the second line is formed as follows from the first. - 1×4=4; 4+-8=-4; this 4 is written under the second term of the first line. -4×4=-16; −16+11=-5; this -5 is written under the third term of the first line. -5×4=-20; -20+-39=-59; this -59 is written under the fourth term. —59×4=—236; −236+31=-205; this -205 is written under the fifth term. -2054-820; -820+-101-921; this -921 is written under the last term. The third line is formed from the second, the fourth from the third, &c., in the same manner. When the calculation is made in this manner, the last terms of the horizontal lines are the coefficients of the transformed equation. 237. 6th, to transform an equation into another which shall want a certain term. +Tx+U=0 (A). Let the equation be "+Pa+QxTM−2. Make x=y+a; y being a new unknown quantity, and a an indeterminate quantity which may be disposed of at pleasure. |