EQUATIONS IX GENERAL,] ALGEBRA 545 x - a ; and since the same mode of reasoning will apply to any equation whatever, the truth of the proposition is evident. We have found that (x - a) (x 3 +p x 2 + qx + r ) = 0; and "as a product becomes = when any one of its factors = 0, therefore the equation will have its conditions fulfilled, not only when x a = 0, but also when X s +p x 2 + q x + r = 0. Let us now suppose that b is a root of this equation; then, by reasoning exactly as before, x 3 --p x i + q x + r = (x 5) (x^ +p"x + j") . By proceeding in the same manner with the quadratic equation x 2 --p"x + q" = 0, we shall find that if c denote one of its roots, then x- +p" x + q" = (x - c) (x + c +p") . So that if we put d = - (c +p"), we at last find # 4 +px z + qx 2 + rx + s = (x - a) (x b) (x - c) (x -d); a, b, c, d, being the roots of the equation, x* +px 3 + qx 2 + rx + s = . The mode of reasoning which has been just now employed in a particular case, may be applied to an equation of any order whatever; we may therefore conclude, that every equation may be considered as the product of as many simple factors as the number denoting its order contains unity, and therefore, that the number of roots in any equa tion is precisely equal to the exponent of the highest power of the unknown quantity contained in that equation. 79. By considering equations of all degrees as formed from the products of factors x - a, x - b, x - c, &c., we dis cover certain relations subsisting between the roots of any equation and its coefficients. Thus, if we limit the num ber of factors to four, and suppose that a, b, c, d, are the roots of this equation of the fourth degree, x 4 +px s + qx 2 + rx + s = , we shall also have (x -a) (x - b) (x - c) (x - d) = ; and therefore, by actual multiplication, + ah + ac - abc ] + ad 9 aid , }-x 2 , -x + abcd=0. + be - acd I +bd -bed } + cd J If we compare together the coefficients of the same powers of x, we find the following series of equations : a+b+c+d= -p , ab + ac + ad + bc + bd + cd +q , abc + abd + acd + bed = r , abcd= + s; and as similar results will be obtained for equations of all degrees, we hence derive the following propositions, which are of great importance in the theory of equations. 1. The coefficient of the second term of any equation, taken with a contrary sign, is equal to the sum of all the roots. 2. The coefficient of the third term is equal to the sum of the products of the roots multiplied together two and two. 3. The coefficient of the fourth term, taken with a con trary sign, is equal to the sum of the roots multiplied toge ther three and three; and so on for the remaining coeffi cients, till we come to the last term of the equation, which is equal to the product of all the roots having their signs changed. Instead of supposing an equation to be produced by mul tiplying together simple equations, we may consider it as formed by the product of equations of any degree, provided tli at the sum of their dimensions be equal to that of the proposed equation. Thus, an equation of the fourth degree may be formed either from a simple and cubic equation, or from two quadratic equations. 80. When the roots of an equation are all positive, its simple factors will have this form, x - a, x b, x - c, <fcc., and if, for the sake of brevity, we take only these three, the cubic equation which results from their product will have this form, x 5 -px^ + qx - r = , where p = a + b + c, q = ab + ac + bc, r = abc ; and here it appears that the signs of the terms are + and - alternately. Hence we infer, that when the roots of an equation are all positive, the signs of its terms are positive and negative alternately. If again the roots of the equation be all negative, and therefore its factors x + a, x + b, x + c, then p, q, and r being as before, the resulting equation will stand thus : x 3 +px 2 + qx + r = . And hence we conclude, that when the roots are all nega tive, there is no change whatever in the signs. In general, if the roots of ail equation be all real, that equation will have as many positive roots as there are changes of the signs from + to - , or from - to + ; and the remaining roots are negative. This rule, however, does nut apply when the equation has impossible roots, unless such roots be considered as either positive or negative. The connection between the signs of the roots and the signs of the terms of an equation can be deduced from the proposition, that the introduction of a new positive root introduces a new change of signs amongst the terms of the equation. The demonstration of this proposition depends on the Rule of fact already established, that an equation may be resolved signs. into the product of simple factors, so that, for example, every equation of the fifth degree may be derived from some equation of the fourth, by multiplying the latter by x - a where a is the additional root. We shall show that the introduction of a new positive root produces an equa tion with at least one more change of signs than the origi nal, and the introduction of a new negative root produces an equation with at least one more continuation of the same sign. To save space, it will suffice if we write the signs without the letters ; thus, x 2 +px - q may be written + H . Let, then, any equation be written down (of the sixth degree, for instance), + + --- 1 --- - ; multiply by x - a, and write the multiplication in the usual form, The signs of the product are all determinate except two, which we have marked with a (?). Now the changes of sign in the original equation are three one between the 1st and 3d terms, one between the 3d and 5th, and one between the 5th and 6th; and it is evident that whatever be the signs marked (T), the produced equation lias as many changes of sign as the original between the same limits, and one change beyond those limits, viz., between the 7th and 8th terms. This proposition is perfectly general, that the introduction of a positive root causes the introduction of at least all the original changes of sign within their limits, and one more change beyond those limits. In the same manner we may prove that the intro duction of a negative root introduces at least one more continuation of the same sign. Hence the conclusion, that an equation cannot have more positive roots than it has changes of sign, nor more negative roots than it has con tinuations of the same sign. This proposition is known as Descartes Rule of Signs. 81. Surd and impossible roots enter equations by pairs. Let a+ ,Jb be a root, where b is a positive or negative number or fraction ; then a - y/> is also a root
I. 69
