546 [EQUATIONS IN GENERAL, If a+ *Jb be written for x in the quantity x n +px" 1 + &c., the result is composed of a series of powers of a and
- Jb. Of these all but the odd powers of *Jb are numerical,
whilst odd powers of *Jb may be written as numerical products of ,Jb itself. The result of the substitution is therefore of the form A + B ^Jb. But since a + Jb is a root of the equation x* +px"^ 1 + &c. = 0, we must have A + B,/& = 0, and.-. A = 0, B = 0. Now if a - sjb be substituted for x, the result will be A - B ,Jb, because even powers of - *]b are the same as those of + Jb. But A = 0, B = .-. A-B ^ = 0; con sequently a fjb is a root of the equation. From this proposition it appears that every equation whose degree is denoted by an odd number, must have at least one real root. Dransfor- 82. We shall now explain some transformations which nation of are frequently necessary to prepare the higher orders of iquations. equations for solution. Any equation may have its positive roots changed into negative roots of the same value, and its negative roots into such as are positive, by changing the signs of the terms alternately, beginning with the second. The truth of this remark will be evident if we take the equation (x -a) (x- b) (x + c) = x 3 +px 2 + qx + r = , and write - x in place of x, producing -(x + a) . -(x + b).(-x + c)= -x 3 +px"* - qx + r = , i.e., (x + a) (x + b) (x-c} = x s -px 2 + qx - r = , where it appears that the signs of the first and third terms are the same as in the original equation, but the signs of the second and fourth are the opposite. And this will be found to hold true of all equations, to whatever order they belong. 83. It will sometimes be useful to transform an equa tion into another that shall have each of its roots greater or less than the corresponding roots of the other equation, by some given quantity. Let (x -a)(x-b)(x + c) = Q be any proposed equation which is to be transformed into another, having its roots greater or less than those of the proposed equation by the given quantity n; then, because the roots of the trans formed equation are to be + a n, + b n, and - c n, the equation itself will be (y =F n - a) (y T n - b) (y T n + c) = . Hence the reason of the following rule is evident. If the new equation is to have its roots greater than those of the proposed equation, for x and its powers substi tute y n and its powers ; but if the roots are to be less, then, for x substitute y + n ; and, in either case, a new equa tion will be produced, the roots of which shall have the property required. 84. By the preceding rule, an equation may be changed into another, which has its roots either all positive or all negative; but it is chiefly used in preparing cubic and bi quadratic equations for solution, by transforming them into others of the same degree, but which want their second term. Let x 3 +2)x 2 + qx + r = be any cubic equation; if we substitute y + n for x, the equation is changed into the following : ) 72 + 3?i 2 +n 3 ] y - + 2pn p y + r Now, that this equation may want its second term, it is evident that we have only to suppose 3n +p = 0, or n = -; for this assumption being made, and the value of n
- j
substituted in the remaining terms, the equation becomes or, putting - + <? = <? , equation may stand thus, + r = r , the same 85. In general, any equation whatever may be trans formed into another, which shall want its second term, by the following rule. Divide the coefficient of the second term of the pro posed equation by the exponent of the first term, and add the quotient, with its sign changed, to a new unknown quantity; the sum being substituted for the unknown quantity in the proposed equation, a new equation will be produced, which will want the second term, as required. By this rule any adfected quadratic equation may be readily resolved; for by transforming it into another equa tion which wants the second term, we thus reduce its solu tion to that of a pure quadratic. Thus, if the quadratic equation # 2 - 5x + 6 = be proposed; by substituting y + f for x, we find _5y_M = 0, ory 2 -i = 0. + 6 ) Hence y = f , and since x = y + ~ , therefore x = | + 1 + 3, or + 2 . ~ 8G. Instead of taking away the second term from an equation, any other term may be made to vanish, by an assumption similar to that which has been employed to take away the second term. Thus, if in Art. 84 we assume 3 2 + 2pn + q = Q, by resolving this quadratic equation, a value of n will be found which, when substituted in the equation, will cause the third term to vanish; and, by the resolution of a cubic equation, the fourth term may be taken away; and so on. 87. Another species of transformation, of use in the resolution of equations, is that by which an equation, hav ing the coefficients of some of its terms expressed by frac tional quantities, is changed into another, the coefficients of which are all integers. Let x 3 +^x^ + -# + -= denote an equation to be so a I) G transformed, and let us assume y = abcx, and therefore x = ; then, by substitution, our equation becomes aoc and multiplying the whole equation by a 3 6 3 c 3 , we have 2/ 3 + bcpy" 2 + cPbfiqy + 3 6 3 cV = . Thus we have an equation free from fractions, while at the same time the coefficient of the highest power of the unknown quantity is unity, as before. Examples of the Transformation and Solution of Equations when certain relations amongst the roots are known. Ex. 1. If a, b, c arc the roots of the equation x s - # 2 + 2x - 3 = 0, to form the equation of which the roots are (1.) a + b , b + c, c + a. Let y be any one root of the required equation; put y = a + b + c-x=-x (Art. 79), and the values of y will be the roots of the equation required, which is therefore or
b+c-a a+c-b a+b-c
