7. The roots of the equation 6x4 43x3 + 107x2 108x b + 36 = 0, are of the form a, b, g, and find them; and shew a' what relation exists between the coefficients of a cubic whose roots are of the form + a, a, and + b. 8. Shew that if an equation have two equal roots, and the terms are multiplied by the terms of an arithmetic progression, the result will = 0. terms may 9. In an equation of n dimensions, the second and third be taken away by the same transformation when the square of the sum of the roots : the sum of their squares ::n:1. Required a proof. 10. The roots of the equation x4 – 10.x3 + 35x2 – 50x + 24 = 0 are of the form a + 1, a-1,6 +1,6-1; find them. 11. Shew that Cardan's rule for the solution of a cubic equation is applicable when all the roots are possible and two of them equal; and by means of it find the roots of the equation x3 + 6x2 32 12. Transform the equation x3 — p.x2 + 2x – y = 0) whose 1823 roots are a, b, c, into one whose roots are = 0. 13. If a be a root of Des Cartes’s reducing cubic, then will 9 4 and + x + V (-4-1) 20 = 14. Take away the third term of the equation x3 6x2 + 9x = 0. 15. In the equation x3 px2 + 2x - y = 0, prove that the sum of the products of the roots and their reciprocals taken three and three together : product of the roots ::1 + p2 + 92 + 72:72 1824 16. If three roots of an equation be nearly equal to one another, and much less than all the others, shew that an approximation may be made to them by the solution of a cubic. 17. Find the sum of the mth powers of the reciprocals of the roots of an equation in terms of the inferior powers. 18. If a be an approximate value of x in any equation, and b, c be the results, when a is substituted for x in the original b с 19. If zon + Axn-1 Prp S + Tx + V 0, where P is the greatest and S the last negative coeffi 1 1 1 1825 6 = rx-3 V8 cient, then is an inferior limit of the positive roots. Vo + P8 20. Solve the equation 23. 6x2 + 11x 0, the roots being in arithmetical progression. 21. Find one of the roots of the equation 3x3 - 26x2 + 34x - 12 = 0 by the method of divisors. 22. The roots of the equation 31 – pxn-1 + 9xn-2 +... = 0 are in geometrical progression beginning from unity; given p = 15,9 = 70. Required n, r, &c. . 23. If the terms of an equation, all whose roots are possible, be multiplied by the terms of the arithmetical progression 0, 1, 2, 3, &c. the resulting equation will be a limiting equation to the former, with this exception, that no root of the limit will lie between the positive and negative roots of the proposed equation. 24. The equation x3 – 7x2 + 16.3 – 12 = 0 has two equal roots. Find all the roots. 25. Transform the equation x3 – px2 + 2x - r = 0, whose roots are a, b, y, into one whose roots are a2 + B2, a2 + y?, and B2 + 7%. 26. Prove that Des Cartes solution of a biquadratic equation succeeds when all the roots are possible and two of them equal, and apply it to solve the equation X4 6x3 + 8x2 + 6x – 9 = 0. 1826 27. If the roots of the equation " – px"-' + qxn-2 - &c. + Qir? - Px + L=0 be in harmonical progression, then will the greatest and least be respectively my m + 1 L ✓n F1 P-V3 (n 1)2 P2 - 6n (n − 1) QL and nvn ( n + 1P - V3 (7 1) P2 – 6n (n -- 1) QL required a proof. 28. In any recurring equation an pxn-1 + 9.2"-2 - &c. + qx? px + 1 = 0, whose roots are a, b, C, &c.; prove that a 62 a? c2 62 + + C2 . (p2 — 29 + vn) (p? — 29 – vn). 29. If an equation has n equal roots, the equation formed 1827 by multiplying the terms by the terms of an arithmetical progression has n - 1 of them. 30. In the equation x4 + 8x3 + x2 -- X – 10 = 0 take away 1828 the second term, and then find the reducing cubic. 31. Find the sum of the sixth powers of the roots of the equation 23 1 - 0. х 32. Shew that Cardan's solution applies only to those cases in which the equation has two impossible roots, unless two of the roots be equal. 33. The coefficient of the second term of an equation with its proper sign, is the sum of the roots with their signs changed; the coefficient of the third term is the sum of the products of every two roots with their signs changed; the coefficient of the fourth term is the sum of the products of every three roots with their signs changed, &c. &c. D - 6 = 0, 1829 34. Shew that the limiting equation has at least as many possible roots as the original equation, wanting one ; and determine the nature of the roots of the equation 27 - ab x2 + c = 0. 23 7x = l. 1 1 to t, as it has positive roots; and as many continuations of the same sign from + to + and from to as it has negative roots. 38. If the equation 23 – px? + 9x — p = 0 has two equal roots, one of them is but the converse is not neces 69 - 2p21 sarily the case. 39. Explain the method of finding those roots of an equation which are whole numbers, by the Method of Divisors, and apply it to solve the equation 9r – pq 23 Arx – 24 9.2 + 40. Shew that any recurring equation of 2m or 2m + 1 dimensions may be solved by an equation of m dimensions. 41. In the solution of a biquadratic by Des Cartes' method, whatever root of the reducing cubic is employed, the same values of the roots of the biquadratic will be obtained. 42. Transform the equation x3 – px? + qux - r = 0, whose roots are a, b, c, into one whose roots are 6 a tc 43. Explain Newton's rule for discovering impossible roots in any equation. 44. In the equation x2 – px +9 = 0, find the sum of the nth powers of the roots in terms of the coefficients. с a + b с a 45. If a, b, c, &c. be the roots of an equation, find the value of a+b + a'c + ba + &c. 46. Find the sum of the mih powers of the roots of an equation in terms of the coefficients and the sums of the inferior powers. 47. Shew the method of extracting the cube root of the binomial surd a + 7 b, and apply it to the solution of the equation x3 - 3x – 18 = 0, by Cardan's rule. 48. Solve by Cardan's rule x3 + 3x2 + 9x – 13 = 0. 49. If two magnitudes, when substituted for the unknown 1830 quantity in an equation, give results affected with different signs, an odd number of roots lies between them ; but if they give results with the same sign, either no root or an even number of roots lies between them. 50. An equation of m dimensions has n equal roots, shew how to find them; and solve the equation x4 + 13x3 + 33x2 + 3x + 10 = 0, which has three equal roots. a a с с a 51. Explain Newton's method of approximating to the roots of an equation, and shew that its accuracy does not depend upon the ratio of the quantity assumed to the root, but upon its being nearer to one root than to any other. 52. The roots of the equation x3 - px? + qx - y = 0 are a, b, and c; transform it into one the roots of which are b 6 + + b a 7 53. Investigate Waring's rule for the solution of a biquadratic equation. 54. Impossible roots enter equations by pairs. 55. Investigate Waring's rule for the solution of a biquadratic equation, and shew that the reducing cubic equation is soluble by Cardan's rule, only in the case where two roots of the given equation are possible and two impossible. 1831 |