4 Of Equations with two Terms. 156. EVERY equation, involving only one power of the unknown quantity, combined with known quantities, may always be reduced to two terms, one of which is made up of all those, which contain the unknown quantity, united in one expression, and the other comprehends all the known quantities collected together. This has been already shown with respect to equations of the second degree, art. 105, and may be easily proved concerning those of any degree whatever. If we have, for example, the equation by bringing all the terms involving x into one member, we obtain freeing x from the quantity, by which it is multiplied, we have x5 = whence we conclude x= In general, every equation with two terms being reduced to taking the root then of the degree m of each member, we have m x= 157. It must be observed, that if the exponent m is an odd number, the radical expression will have only one sign, which will be that of the original quantity (131). When the exponent m is even, the radical expression will have the double sign ±; it will in this case be imaginary, if the quan tity is negative, and the question will be absurd, like those of Р which we have seen examples in equations of the second degree (131). See some examples. The equation x5 = 1024, leads only to imaginary values, because while the exponent 4 is even, the quantity under the radical sign is negative. 158. I shall here notice an analytical fact, which deserves attention on account of its utility, as well in the remaining part of the present treatise, as in the Supplement, and which is sufficiently remarkable in itself; it is this, that all the, expressions x — ɑ, x2 — ɑ2, x3 — a3, and in general xam (m being any positive whole number), are exactly divisible by x-a. This is obvious with respect to the first. We know that the second 3 . and the others may be easily decomposed by division. If we divide am am by x a, we obtain for a quotient x2-1 + α xm-2 + a2 x3-3 + &c. the exponent of x, in each term, being less by unity than in the preceding, and that of a increasing in the same ratio. But instead of pursuing the operation through its several steps, I shall present immediately to the view the equation =xm-+α xm-2 + a2 xm-3 +am-2x + am-1, ... • which may be verified by multiplying the second member by x-α. It then becomes all the terms in the upper line, after the first, being the same, with the exception of the signs, as those preceding the last in the lower line, there only remains after reduction, am — ɑm, · am, that is, the dividend proposed. It must be observed, that the term a2 xm-2, in the upper line, is necessarily followed by the term a3 xm-3, which is destroyed by the corresponding term in the lower line; and that, in the same manner we find, in the lower line, before the term am-1x, a term — am-2x2, which destroys the corresponding one in the upper line. These terms are not expressed, but are supposed to be comprehended in the interval denoted by the points. 159. This leads to very important consequences, relative to the equation with two terms am Р If we designate by a the number, which is obtained by directly extracting the root according to the rules given in art. 154, we have The quantity — am is divisible by x-a, and we have am the preceding article xm am = (x ·α) (xm-1 + α xm−2 ..... by + am-2 x + am−1). This last result, which vanishes when xa, is also reduced to nothing, if we have ..... xm−1 + a xm-2 + am-3x + am-10. (116); and, consequently, if there exists a value of x, which satisfies this last equation, it will satisfy also the equation proposed. These values have with unity very simple relations, which may be discovered by making xay; then the equation 0 becomes am ym — am = 0, or y"—1 = 0, and we obtain the values of x, by multiplying those of y by the number a. The equation y — 1 = 0, gives in the first place yTM = 1, y = ŵi = 1; then by dividing y 1 by y — 1, we have Taking this quotient for one of the members, and zero for the other, we form the equation on which the other values of y depend; and these values will, in the same manner, satisfy the equation ym. 1 = 0, or ym = that is, their power of the degree m will be unity. Hence we infer the fact, singular at first view, that unity may have many roots beside itself. These roots, though imaginary, are still of frequent use in analysis. I can, however, exhibit here only those of the four first degrees, as it is only for these degrees, that we can resolve, by preceding observations, the equation The last two are imaginary; but if we take the cube, forming that of the numerator, by the rule given in art. 34, and observing that the square of 3 being - 3, its cube is -3, its cube is 33, we still find y3 = 1, in the same manner as when we employ the root y = 1. 3. Taking m = 4, we have y' -10, from which we deduce then y= 1, y3 + y2 + y + 1 = 0. We are not, at present, furnished with the means of resolving this equation; but observing that y1 — 1 = (y2 + 1) (y2 — 1), we have successively whence y210, y2 + 1 = 0, y = +1, y = 1, y = + √=1, y=√1. Two of these values only are real; and the other two imaginary. This multiplicity of roots of unity is agreeable to a general law of equations, according to which any unknown quantity admits of as many values, as there are units in the exponent denoting the degree of the equation, by which this unknown quantity is determined; and when the question does not admit of so many real solutions, the number is completed by purely algebraic symbols, which being subjected to the operations, that are indicated, verify the equation. Hence it follows, that there are two kinds of expressions or values for the roots of numbers; the first, which we shall term the arithmetical determination, is the number which is found by the methods explained in art. 154, and which answers to each particular case; the second comprehends negative values and imaginary expressions, which we shall designate by the term algebraic determinations, because they consist merely in the combination of algebraic signs. Of Equations which may be resolved in the same manner as those of the Second Degree. 160. THESE are equations, which contain only two different powers of the unknown quantity, the exponent of one of which is double that of the other. Their general formula is x2m + pxm = 4, p and q being known quantities. Now if we take am for the unknown quantity, and make mu, we have whence x2m = u, u2 + pu= 9, |