.... integer, and A, B, C, ........ L, M, N, any numbers whatever, either positive or negative, or zero, but all free from the imaginary symbol. AxBx-1+ Сx2-2 + .... + Lx2 + Mx + N = 0. Some classes of reasonings, however, are facilitated by having unity for the coefficient of x", and the equation is at once reduced to this form by division of all the terms by A. 2. For brevity of writing, this is often put in the contracted form The symbols ƒ (x) and X are in this case called functions of x; meaning an expression into the composition of which a enters, or which depends upon the value of x. 3. A root of an equation is any number or expression which, on being substituted in the given equation, and all reductions being performed, fulfils the expressed condition of making both sides equal by the mutual cancel of all the terms on the first side. .... 4. If r1, T2, T3, ... be roots of an algebraic equation ƒ (x) = 0, then the successive quotients of f (x) by x-r1, x—r2, are called the depressed equations. It is seldom that these depressed equations require any special notation; but when they do, 4, (x) = 0, $2 (x) = 0, $3 (x) = 0 ... are found convenient. 5. If f (x) be divided by x-x, these quotients are called the first, second, third,.... derivative functions of x, or simply derivatives. They are respectively denoted by ƒ, (x), ƒ2(x), ƒ2(x), · f(x) according to the number of successive divisions performed. .... 6. An equation is said to be transformed when it is changed into another whose roots have any assigned relation to those of the given one: as, for instance, when an equation is given, and another is formed from it whose roots shall be triple, one half, or any multiple or part of those of the given one; or again when the roots shall be all greater or less than those of the given equation by some given quantity; and so on. In the last-mentioned case, the new equation is called the reduced equation. 7. By a permanence of signs, or simply a permanence, is meant that two consecutive terms of a complete equation have the same sign prefixed; as + + or and by a variation, that two consecutive terms have unlike signs prefixed, as + - or +. .... THEOREM 1. If r, r1, r1⁄2, . . . . be roots of an equation, ƒ (x) = 0, then ƒ (x) is exactly divisible by x-r, x-r1, X—T29 .... .... without remainder. Let f (x) = Ax” + Bx”-1 + Cx"−2 + +Ma+N=0 be the given equation, and let the first side be divided by x-r by the synthetic method, p. 128. where A, B,, C1, .......... L1, M,, N, are the coefficients of the quotient, as far as .... the term æ ̄1; and we have to show that N, = 0, and hence that the quotient terminates at M1. By attending to the formation of the coefficients of the quotient, we see that M1 = L1r + M = Ap2¬1 + Bpr−2 + Cpr−3 + N1 = M1r + N = Arr + Bp"-1+Cp-2 + But by hypothesis r is a root of the equation f (x) it for a we have .... + Lr + M,. + Lr2 + Mr + N. = 0; hence substituting + Lr2 + Mr + N = 0, ATM” + Вp”−1 + Сpr~2 + and as this is exactly the value of N, found above we have N1 = 0, and the division terminates. 1 In the same manner it may be shown that ƒ (x) is divisible by x-r¡, X—T2, . . Cor. Since f(x) is divisible separately by x-r, x-r1, x-r2,.. it is also divisible by their product. THEOREM II. The derivatives may be formed by inspection in the following manner: Multiply each term by the index of the power of x in it, and diminish that index by unity, which will give the first derivative: operate upon the first derivative in the same manner to produce the second; upon the second to produce the third; and so on to the end. For the division by x-x gives the same coefficients in terms of x that the division by x-r gives in terms of r; hence restoring a for r in the values of B1, C1,.. L1, M1, we have The second derivative ƒ, (x) will obviously be found from this in the same manner, since the same reasoning applies to all the derivatives in succession. Hence the several derivatives are f(x)= nАx-1 + (n−1) Bá”—2 + (n−2) Сx2-3 + .... f2 (x) = n (n−1) Ax”−2 + (n−1) (n−2) Bx”−3 + (n−2) (n−3) Cx2+ .... fn-2 (x) = n (n−1)..........3Ax2 + (n−1) (n−2)........2Bx + (n−2) (n−3)....2.1 Jan () = n - -1)....2Ax + (n−1) (n−2)....1.B 2-1 Jn (@) = n(n−1)....2.1.A PROBLEM I. To calculate the value of an algebraical function when the value of x is given; that is, to find the value of Ax" + Bæ”-1 +........ + Mx + N, when x is a given number, the coefficients being also given numerically. Range the coefficients in a horizontal line, as for synthetic division, with their proper signs prefixed, taking care when any coefficients are absent to fill their places with ciphers. Multiply A by the given value of x, and add the result to B, making the sum B,; multiply B, by a and add the result to C giving C1, and so on till we arrive at N1. Then N, is the value of the expression. For this is precisely the operation performed in theorem 1; and by the reasoning of that theorem and we were required to find its value when x = 5 and when x = — work would stand thus: 2, then the 2095481 + 2409 + 12035 + 60190 = value when x = 5. - 8+ 11 1636 82179 value when a = — 2. To transform an algebraic equation, having its roots less by a given quantity a than the roots of the given equation. Divide the given function continually, employing the synthetic method, by x-a, always stopping at the term where (x-a)-1 occurs: the several terminal quotients will be the coefficients of the reduced equation. The following is its general type. * Let Ax" + Bx"−1 + Сx”−2 + ... Lx2 + Mx + N = 0 be the given equation: then, dividing synthetically, we have the several operations as follow: + K + L +M+N +H,a+Ka+La+M1a + K2 + L2 + M2 + K2+ L3 A (x−a)”+ B„ (x—a)”—1+Сn-1 (x—a)”—2+.......... L ̧ (x—a)2+M, (x—a)+N1=0. .... * It will be more convenient to place the transforming number to the right of the coefficients, separated by the curve employed in division or extraction of roots, as in the numerical illustrations which follow. For the results of the successive divisions performed as above, are Cor. 1. If we diminish the roots of an equation by a quantity a, which is greater than Ρ of the positive roots a, a, ɑ2, .ap, then in the reduced equation, p of the positive roots corresponding to these will be negative, viz. a .... ... .... ap, α1 ap ap-1 ap, all of which are negative, since a, is the greatest of all the quantities. In a similar manner, if the roots of an equation are increased by a quantity a, which is greater than a, a, ap-1, (―a, ap-1, being p negative roots,) then the p roots of the reduced equation corresponding to these will become positive. Of course the transformed roots corresponding to those which were negative before the diminution, or positive before the increase of the roots, retain still the same character as their original corresponding roots. Cor. 2. Had we been required to form an equation whose roots are greater than those of a given equation, the process would obviously have been the same, only dividing continually by x + a instead of x a; that is, by using the factor a instead of a used above in forming the several successive courses of coefficients. As examples, let 2x4. 10x3 +20x2 15x+100 be transformed into an equation whose roots are less by 4, and the equation 2y1 + 22y3 + 92y2 + 177y + 142 = 0 into one whose roots shall be greater by 4. The work will stand thus:: *Other methods requiring rather less work but involving principles rather less comprehensible by the student, (and given, too, without investigation,) may be seen in Leybourn's Repository, vol. v. pp. 42--44. The demonstrations of them will appear in my forthcoming publication of Mr. Horner's works on Equations.-EDITOR. It will at once appear that the second given equation is but a transformation of the first, and vice versa. The restoration of the original result proves the truth of the transformation and re-transformation. When the transformation is to be made by a number comprising more than one figure, it may first be transformed by the first of them (regard being had to its place in the decimal scale), then this transformed equation again transformed by the next figure, then again this by the next figure, to any assigned extent, the same precaution respecting decimal place being observed in all. Thus, to reduce the roots of the equation 3 + 3x2 + 3x 1400 by 4.23, we shall have *392 L03 2'447667 (x −4·23)3 + 15·69 (x — 4·23)2 + 82·0587 (x — 4·23) + 2·055667 = 0 *. But in many cases, especially where the process does involve much intermixture of the signs + and · the whole work may be more advantageously per * Though the decimal points are marked in this process, they will, after a little practice, be easily dispensed with by the pupil, as the regular arrangement and increase of the places to the right will always secure the figures falling rightly. |