If the last of these equations has a root, c, its first member may still be decomposed into two factors, we then have x — c, x2-3 + P x + &c. = 0; from which it is obvious, that the proposed equation may be veri fied in four ways, namely, by making X- — a = 0, x — b = 0, x — - − c = 0, x2¬3 + Plan + &c. = 0. Pursuing the same reasoning, we obtain successively factors of the degrees N ·4, n-5, n-6, &c.; and if each of these factors being put equal to zero, is susceptible of a root, the first member of the proposed equation is reduced to the form 1), (x — a) (x — b) (x — c) (x — d) . (x that is, it is decomposed into as many factors of the first degree, as there are units in the exponent, n, which denotes the degree of the equation. The equation x2 + Pα-1 + &c. = 0, may be verified in n ways, namely, by making x-α=0, or x-b0, or x-c=0, or x―d=0, a = 0, It is necessary to observe, that these equations are to be regarded as true only when taken one after the other, and there arise manifest contradictions from the supposition, that they are true at the same time. In fact, from the equation x we obtain xa, while x b = 0 gives xb, results, which are inconsistent, when a and b are unequal quantities. 182. If the first member of the proposed equation, x2 + Px2-1 + &c. = 0, be decomposed into n factors of the first degree, it cannnot be divided by any other expression of this degree. Indeed, if it were possible to divide it by a binomial x — α, different from the former ones, we should have x2 + P x2-1 + &c. = (x — α) (xn−1 + px2-2 + &c.) and, consequently, = (x — α) (xn−1 + pxn-2 + &c.); now by changing a into a, this becomes (aa) (ab) (a — c) (α — d) .. - (x-1) (α — 1) The second member vanishes by means of the factor aα, which is nothing; this is not the case with respect to the first, which is the product of factors, all of which are different from zero, so long as a differs from the several roots a, b, c, d. 1. The supposition we have made then is not true; therefore, an equation of any degree whatever does not admit of more binomial divisors of the first degree, than there are units in the exponent denoting its degree, and consequently, cannot have a greater number of roots.* 183. An equation regarded as the product of a number of factors, equal to the exponent of its degree, may take the form of the product exhibited in art. 135, with this modification, that the terms will be alternately positive and negative. If we take four factors, for example, we have bx3 + ac x2 cx3 + ad x2 - d x3 + b c x2 + cdx2 The second terms of the binomials x α, x - b, X- c, &c. being the roots of the equation, taken with the contrary sign, the properties enumerated in art. 135, and proved generally in art. 136, will, in the present case, be as follows, The coefficient of the second term, taken with the contrary sign, will be the sum of the roots; The coefficient of the third term will be the sum of the products of the roots, taken two and two; The coefficient of the fourth term, taken with the contrary sign, will be the sum of the products of the roots, multiplied three and * This demonstration is taken from the Annales de Mathématiques published by M. Gergonne. See vol. iv, pp. 209, 210, note. three, and so on, the signs of the coefficients of the even terms being changed; The last term, subject also to this law, will be the product of all the roots. Making, for example, the product of the three factors for the sum of their products, taken two and two, +5X-4+5X-3-4X-3-2015 +12=— 23, and for the product of the three roots, In this way we form the coefficients, 2, — 23, — the signs of those for the second and fourth terms. If we make the product of the factors equal to zero, the equation thence arising X3 19x+30= 0, 60, changing as it has no term involving x2, the power immediately inferior to that of the first term, wants the second term; and the reason is, that the sum of the roots, which, taken with the contrary sign, forms the coefficient of this term, is here or zero, or in other words, the sum of the positive roots is equal to that of the negative.* 184. We have proved (182), that an equation, considered as arising from the product of several simple factors, or factors of the first degree, can contain only as many of these factors, as there are units in the exponent n denoting the degree of this equation; but if we combine these factors two and two, we form quantities of the second degree, which will also be factors of the proposed equation, the number of which will be expressed by (x — a) × (x —b) × (x — c) × (x — d), may be decomposed into factors of the second degree, in the six following ways; (x — a) (x b) × (x — c) (x — d) By combining the simple factors three and three, we form quantities of the third degree for divisors of the proposed equation; for an equation of the degree n the number will be Of Elimination among Equations exceeding the First Degree. 185. THE rule given in art. 78, or the method pointed out in art. 84, is sufficient, in all cases, for eliminating in two equations an unknown quantity, which does not exceed the first degree, whatever may be the degree of the others; and the rule of art. 78, is applicable, even when the unknown quantity is of the first degree in only one of the proposed equations. If we have, for example, the equations а x2 + bxy + cy2 = m2, x2 + xy = n2, taking, in the second, the value of y, which will be and substituting this value and its square, in the place of y3 in the first equation, we obtain a result involving only x. y and 186. If both of the proposed equations involved the second power of each of the two unknown quantities, the above method could be applied in resolving only one of the equations, either with respect to x or y. Let there be, for example, the equations the second gives a x2 + bxy + cy2 = m2, x2 + y2 = n2; Substituting this value of y, and its square in the first, we obtain а x2 + bx √n2 x2 + c (n2 x2) = m2. Our purpose appears to be answered, since we have arrived at a result, which does not involve the unknown quantity y, but we are unable to resolve the equation containing x, without reducing it to a rational form, by making the radical sign, under which the unknown quantity is found, to disappear. It will be readily seen, that if this radical expression stood alone in one member, we might make the radical sign to disappear by raising this member to a square. Collecting together all the rational terms then in one member, by transposing the termsbx vn2 x2 and m2, we have a x2 + c (n2 -X x2) m2 = b x √ ñ taking the square of each member, we form the equation 4 a2x2+c2 (n2 —x2)2+m1 +2acx2(n2x2)—2am2x2-2cm2 (n2—x2 which contains no radical expression. ⇒b2x2 (n2 —x2), The method, we have just employed for making the radical sign to disappear, deserves attention, on account of the frequent occasion we have to apply it; it consists in insulating the quantity found under the radical sign, and then raising the two members of the proposed equation to the power denoted by the degree of this sign. 187. The complicated nature of this process, which increases in proportion to the number of radical expressions, added to the difficulty of resolving one of the proposed equations with reference to one of the unknown quantities, a difficulty, which is often insurmountable in the present state of algebra, has led those, who have cultivated this science, to seek a method of effecting the |
