Solution of Equations of a peculiar Form. 1 ( 2 2 + 1 ) + B ( x + 1 ) + C = 0, and, if we make x 2 which are to be substituted in the values of x, x = + y = √(† y2 — 1), x6+3x57x46x37x2 + 3x + 1 = 0. Solution. Divide by z3, and we have Values of Coefficients in Equations. and the equation becomes, by substitution, y3+3 y2-10y= 0. The roots of this equation are y = 0, 2, and = 5; = and, therefore, the values of x are x = ± √ − 1, = 1, or = √(−5±√21). 3. Solve the equation x8+2x6 6x + 2x2 + 1 = 0. Ans. x = ± 1, or = ±†√2 (2 ± √− 1). 166. Corollary. It follows, from art. 163, that an equation of the second degree has two roots, both of which are given by the process of art. 154; and if the equation is reduced to the form x2 + ax + b = 0, and the roots denoted by x' and x", we have x2 + a x + b = (x — x') (x — x") = 0. But the product (x — x') (x — x") being arranged according to powers of x, is x2 x2 - (x' + x') x + x'x"; which being compared with its equal that is, the coefficient of x is the negative of the sum of the roots of equation, and the term which does not contain is the product of the roots. Values of Coefficients in Equations. 167. Corollary. If the roots of the general equation of the third degree are denoted by we have x 3 + a x2 + bx + c = 0. x', x', x'"'; x3+ax2+bx+c = (x — x') (x — x'') (x — x''') = 0. But the product (x — x') (xx) (x — x'"') is, when arranged according to powers of x, x3 — (x'+x'"'+x''') x2 + (x' x'' + x' x'"'+x'' x''') x —x' x'' x''' ; whence, by comparison with the given equation, we have (x2 + x + x'""'), b = x' x'' + x' x'"' + x'' x'"', Ꮖ that is, the coefficient of x2 is the negative of the sum of the roots, the coefficient of x is the sum of the products of the roots multiplied together two and two, and the term which does not contain x is the negative of the continued product of the roots. 168. Corollary. It may be shown in the same way that, in the equation 3 xn + a xn−1 + b xn−2 + c xn−3 + &c. = 0, the coefficient of x-is the negative of the sum of the roots; the coefficient of x-2 is the sum of the products of the roots multiplied together two and two; the coefficient of x-3 is the negative of the sum of the ducts of the roots multiplied together three and three; and so on, the last term being the product of the roots when n is even, and the negative of this product when n is odd. pra All the Roots of an Equation diminished by the same Quantity. SECTION II. Transformation of Equations." 169. Problem. To transform a given equation into another in which the roots are all diminished by the same quantity. Solution. Let the given equation be x2 + a x2¬1 + bxn−2+ &c. += = 0; the roots of which are x', x", x'", &c., and let e be the excess of these roots above those of the required equation, which must consequently be x' e, x" e, x'!! e, &c.; or if u is the unknown quantity of this new equation, we have x — eu, and this value of x, being substituted in the given equation, produces the required equation, or (e+un+a (e+u)n−1 +b (e+u)n−2+ &c. = which, being arranged according to powers of u, is un + ne\un-1..+ &c... + n en—1 + a 0; |u + en = 0. -2 +ben-2 &c. + (n-1) a en. +aen-1 + (n-2) ben-3 + &c. 170.- Corollary. Since e is now entirely arbitrary, it may be taken to satisfy any proposed condition, such for instance as that the coefficient of un— 1 may be equal to zero, in which case the second term vanishes, and the equation becomes of the form un + b'un - 2 + c2 un-3 + &c... = 0. All the Roots of an Equation diminished by the same Quantity. This condition is represented by the equation ne + a = 0, which gives for the value of e, α e n 171. Corollary. Since the roots of the equation just attained are they become, when e is taken, equal to one of the roots x', x', 'x'"', &c. of the proposed equation, such as x for instance, and, therefore, one of the roots of the equation thus obtained must be zero. Now this equation is which must satisfy it, its first member is reduced to the last term, and we have x1n + ax'n−1 + b x n − 2 + &c. = 0; which equation is evidently correct, since it only differs from the given equation by the substitution for x, of one of its roots x'. Hence the above equation, divided by u, is reduced to un−1 + n x'\un -2 + &c... + n x′ n − 1 = 0. +(n−1) ax'n― 2 + &c. |