Now we know the equation whose roots are less by 1 than those of the given equation it is x+x+2x-2=0; and by a similar process for 7, remembering the localities of the decimals, we have the required equation; thus: ••• y3+3·1y2+4·87y+•233=0 is the required equation. This latter operation can be continued from the former without arranging the coefficients anew in a horizontal line, recourse being had to this second operation merely to show the several steps in the transformation, and to point out the equations at each step of the successive diminutions of the roots. Combining these two operations, then, we have the subsequent arrange We have then the same resulting equation as before, and in the latter of these we have used 17 at once. It is always better, however, to reduce continuously as in the former, to avoid mistakes incident to the multiplier 1·7. (4) Find the equation whose roots shall be less by 1 than those of the equation x3-7x+7=0. (5) Find the equation whose roots shall be less by 3 than the roots of the equation x-3x3-15x+49x-12=0, and transform the resulting equation into another whose roots shall be greater by 4. (6) Give the equation whose roots shall be less by 10 than the roots of the equation x2+2x3+3x2+4x-12340=0. (7) Give the equation whose roots shall be less by 2 than those of the equation x+2x3 — 6.x2-10x+8=0. (8) Give the equation whose roots shall each be less by than the roots of the equation 252. If the real roots of an equation, taken in the order of their magnitudes, be where a is the greatest, a, the next, and so on; then if a series of numbers, in which b1 is greater than a1, b, a number between a, and ɑ, b ̧ a number between ag and a, and so on, be substituted for x in the proposed equation, the results will be alternately positive and negative. The polynomial in the first member of the proposed equation is the product of the simple factors (x—a ̧)(x — a2)(x —α3)(x —α4) and quadratic factors, involving the imaginary roots; but the quadratic factors have always a positive value for every real value of x (Art. 248, Cor. 3); therefore we may omit these positive factors; and substituting for r the proposed series of values, b1, b2, b3, &c., we have these results: Corollary 1.—If two numbers be successively substituted for x in any equation, and give results with different signs, then between these numbers there must be one, three, five, or some odd number of roots. Corollary 2.-If the results of the substitution in corollary 1 are affected with like signs, then between these numbers there must be two, four, or some even number of roots, or no root between these numbers. Corollary 3.-If any quantity q, and every quantity greater than q, renders the result positive, then q is greater than the greatest root of the equation. Corollary 4.-Hence, if the signs of the alternate terms be changed, and if p, and every quantity greater than p, renders the result positive, then -p is less than the least root. EXAMPLE. Find the initial figure in one of the roots of the equation 23-4x2-6x+8=0. Here one value of r does not differ greatly from unity, for the value of the given polynomial, when x=1, is -1, and when x= =9, it is found thus: 1-4-6 +8 (9 -3.1-8.79 ·089. The value, therefore, when r=·9 is (Art. 251) 089. Hence the former value being negative, and the latter positive, the initial figure of one root is ⚫9. PROPOSITION. 253. Given an equation of the nth degree to determine another of the (n—1)* degree, such that the real roots of the former shall separate those of the latter. Let a,, A2, A3, A4, .... an be the roots taken in order of the equation x+A,1+A2+.... A2-1+A2=0; then diminishing the roots of this equation by r (Art. 251), we have the following process, viz.: 1+A‚+ A2+........... A+ A-1+ A。 († ..... (r TB TB 2 TB-1 Whence T rB1 n-3 C-1A-1+ ↑ B2-2+ r Сn-2 =An-1+ 7(An-2+ r Ba-3)+ r (An-2 +r B2-3+rCa-3) =An-1+2r An-2+2r2 Bn-3 + 12 Cn-3 An-1+2r An-2+2r2(A-3 + r B2¬)+r2(A‚¬з+rB1¬+rCo→) Ca-nr-1+(n−1)A1r2+(n−2) Agra-3+... 2A-27+A-1... (1) Again, the roots of the transformed equation will evidently be a1-r, as-r, Aз—r, A4—r, ... an―r, and as we have found the coefficient, C-1, of the last term but one in the transformed equation, by one process, we shall now find the same coefficient, C-1, by another process (Prop. V., p. 309); it is the product of every (n-1) roots of the equation (1) with their signs changed; hence we have Now these two expressions which we have obtained for C-1 are equal to one another, and, therefore, whatever changes arise by substitution in the one, the same changes will be produced, by a like substitution, in the other; hence, substituting a1, a2, a3, &c., successively for r in the second member of equation (2), we have these results: But when a series of quantities, a1, a2, aз, α, &c., are substituted for the unknown quantity in any equation, and give results which are alternately + and —, then, by Art. 352, these quantities, taken in order, are situated in the successive intervals of the real roots of the proposed equation; hence, making C-1=0, and changing r into x, we have from equation (1) nx"1+(n−1)A ̧TM2¬2+(n−2)  ̧Ãa¬3+.... 2 An-22+A-1=0... (3) an equation whose roots, therefore, separate those of the original equation +A1+A2+....An-12+A=0, and the manner of deriving it from the proposed equation is to multiply each term of the proposed equation by the exponent of x, and to diminish the exponent one. It is identical with the second column of the development in the note to Article 251. It is known by the name of the derived equa tion. Let α1, A2, A3, A4, &c., be the roots of the proposed equation, and b1, b1⁄2, b3, &c., those of the derived equation (3), ranged in the order of magnitude; then the roots of both the given, and the derived equation will be represented in order of magnitude by the following arrangement, viz. : a1, b1, a2, b2, aз, bз, ɑ4, b4, ɑ5, b5, &c. . . Corollary 1.-If a2=a1, then r-a, will be found as a factor in each of the groups of factors in equation (2), which has been shown to be the separating equation (3), and, therefore, the separating equation and the original equation will obviously have a common measure of the form x-a1. Corollary 2.-If aз=α=a1, then (r—a1)(r—a1) will occur as a common factor in each group of factors in (2); that is, the separating equation (3) is divisible by (x—a1)2; and, therefore, the proposed equation and the separating equation have a common measure of the form (x—a1)2. Corollary 3.-If the proposed equation have also a=a5, then it will have a common measure with the separating equation of the form (x—a1)2 (x—a1), and so on. Scholium. When, therefore, we wish to ascertain whether a proposed equation has equal roots, we must first find the separating equation, and then find the greatest common measure of the polynomials constituting the first members of these two equations. If the greatest common measure be of the form (x—α1)3 (x —α2)a (x—A3)' then the proposed equation will have (p+1) roots a1, (q+1) roots =a2, (r+1) roots =a3, &c. The equation may then be depressed to another of lower dimensions, by dividing it by the difference between r and the repeated root raised to a power of the degree expressed by the number of times it is repeated. X EXAMPLES. Find the equal roots of the equation x2+5x+6x5—6x1—15x3—3x2+8x+4=0 The derived polynomial is 7x+30x+30x*—24x3-45x2-6x+8. and the common divisor of (1) and (2) x2+3x3+x2-3x-2. The values of x, found by putting this equal to zero, would be the repeated roots of the proposed equation. This itself will be found to have equal roots, for its derived is and by the rule in the above scholium the given equation may be put under the form (x+1)3 (x-1)2 (x+2)3, so that in the proposed equation there are three roots equal to +1, and two to -2. so that the three roots are two equal to -a, and the third 2a. (x-1) (x-2) (x+1)2 (x-3)3=0. 254. The most satisfactory and unfailing criterion for the determination of the number of imaginary roots in any equation is furnished by the admirable theorem of Sturm, which gives the precise number of real roots, and, consequently, the exact number of imaginary ones, since both the real and imaginary roots are together equal to the number denoted by the degree of the proposed equation. PROPOSITION. To find the number of real and imaginary roots in any proposed equation. The acknowledged difficulty which has hitherto been experienced in the important problem of the separation of the real and imaginary roots of any proposed equation is now completely removed by the recent valuable researches of the celebrated Sturm; and we shall now give the demonstration of the theorem by which this desirable object has been so fully accomplish |