EquaTroxs, nature of. Any equation involving the powers of one unknown quantity may be reduced to the form p q = 0, here the whole expression is equal to nothing, and the terms are arranged according to the di mensions of the unknown quantity, the coefficient of the highest dimension is unity, understood, and the coefficients p, q, r, and are effected with the proper signs. An equation, where the index is of the highest power of the unknown quantity is n, is said to be of n dimen sions, and in speaking simply of an equa tion of n dimensions, we understand one reduced to the above form. Any quan tity p q P z—Q may be supposed to arise from the multi plication of z—a X z—b X z— c, &c to n factors. For by actually multiply ing the factors together, we obtain a quantity of n dimensions similar to the proposed quantity an— p q zn—", &c. ; and if a, b, c, &c. can be so assum ed, that the coefficients of the corres ponding terms in the two quantities be come equal, the whole expressions coin cide. And these coefficients may be made equal, because these will be n equations, to determine n quantities, a, b, c, &c. If then the quantities, a, b, c, &c. be pro perly assumed, the equation za—p zn—H q = 0, is the same with z — a Xz—bXz—c, &c. = 0. The quanti ties a, 5, c, d, &c. are called roots of the equation, or values of z ; because, if any one of them be substituted for z, the whole expression becomes nothing, which is the condition proposed by the equa tion.
Every equation has as many roots as it has dimensions. If p +p gcc. = orz—axz—b X z--c, &c.
to n factors = 0, there are n quantities, a, b, c, &c. each of which when substi tuted for z makes the whole = 0, because in each case one of the factors becomes = 0 ; but any given quantity different from these, as e when substituted for z, gives the product e— a X e—b X e—c, &c. which does not vanish, because none of the factors vanish, that is, e will not answer the condition which the equation requires.
When one of the roots, a, is obtained, the equation z— a X &c.
zn—p 4- q &c. = 0 is di visible by z — a without a remainder, and is thus reducible to z—b X z— c, &c. — 0, an equation one dimension low er, whose roots are b and c.
Ex. One root of x3 + 1 = 0, or x 1 = 0, and the equation may be de pressed to a quadratic in the following manner : +1)53+1(x'—x+1 53-Fx' —5' + x+1 x+1 • Hence the other two roots are the roots of the quadratic x' — x 1 = 0. If two roots, a and b, be obtained, the equa tion is divisible by x—a X x b, and may be reduced in the same manner two dimensions lower.
Ex. Two roots of the equation — 1 = 0, are + 1 and —1, or z —1= 0, and z 1 = 0; therefore it may be depress ed to a biquadratic by dividing by z — 1 z' —I.
I +V.
24-2' +? z'—1 • • Hence the equation + 2' +1 =0 con tains the other four roots of the proposed equation.
Conversely, if the equation be divisi ble by x — a without a remainder, a is a root; if by x—a a and b are both roots. Let Q be the quotient aris ing from the division, then the equation isx—aXx—b X Q= 0, in which, if a or b be substituted for x, the whole vanishes.