552 ALGEBRA [EQUATIONS WITH + rx + s = have two equal roots, each of them must also be a root of this equation, for when the first of these equations is divisible by (x a) 2 , the latter is necessarily divisible by x a. Let us next suppose that the proposed equation has three equal roots, or a = b = c ; then, two at least of the three equal factors x - a, x b, x c, must enter each of the four products A, A , A", A ", it is evident that A + A + A" + A ", or Ix 3 + 3px 2 + 2qx + r must be twice divisible by x - a. Hence it follows that as often as the proposed equation has three equal roots, two of them must also be equal roots of the equation ^x 3 + 3px- + 2qx + r = Q. 109. Proceeding in the same manner, it may be shown, that whatever number of equal roots are in the proposed equation x* +PX 3 + qx 2 + rx + s = , they will remain, except one, in this equation 4.C 3 + 3px 2 + 2qx + r = 0, which may be derived from the former, by multiplying each of its terms by the exponent of x in that term, and then diminishing the exponent by unity. 110. Tf we suppose that the proposed equation has two equal roots, or a b, and also two other equal roots, or c = c/, then, by reasoning as before, it will appear that the equa tion derived from it must have one root equal to a or /<, and another equal to c or d ; so that when the former is divisible both by (x a) 2 and (x c) 2 , the latter will be divisible by ( x a) (x- c). 111. As a particular example, let us take this equation, x 5 -13r* + G7^ 3 - 171^ 2 + 216.r -108 = 0, and apply to it the method we have explained, in order to discover whether it has equal roots, and if so, what they are. We must seek the greatest common measure of the proposed equation and this other equation, which is formed agreeably to what has been shown (Art. 109), 5x* - 52x* + 201.r 2 - 32x + 21G = ; and the operation being performed, we find that they have a common divisor, x 3 8x 2 + 2lx- 18, which is of the third degree, and consequently may have several factors. Let us therefore try whether the last equation, and the following, 20x* - 156.r 2 + 402x - 342 = , which is derived from it by the same process, have any common divisor ; and, by proceeding as before, we find that they admit of this divisor, x - 3, which is also a factor of the last divisor, x 3 - 8.r 2 + 21 x - IS ; and therefore the product of the remaining factors is immediately found by division to be x~ 5x + G, which is evidently resolvable into x - 2 arid x 3 . Thus it appears that the common divisor of the original equation, and that which is immediately derived from it, is (x-2) (x-3) 2 ; and that the common divisor of the second and third equations is x - 3. Hence it follows that the proposed equation has (x - 2) 2 for one factor, and (x 3) 3 for another factor, and may therefore be expressed thus, (x - 2) 2 (x - 3) s = 0. The truth of this conclusion may be easily verified by multiplication. The five roots are 2, 2, 3, 3, and 3. 112. The property proved in Art. 107 enables us to establish numerous relations between the coefficients and roots of an equation, in addition to the fundamental one established in Art. 79, such as the following: Since x* +px*~ 1 + qx*~ 2 + ike. = (x - a) (x - b) (x - c) &c. and nx"~ l + (n - 1 )px*~ 2 + (n- 2)^"- + (fee. = (x - l)(x -c). . . +(x- a)(x - c) ... + ..., by division 71X"" 1 + (x - l)px"~* + (n - 2)qx- 3 + &c.
- b x c
where S 15 S 2 , S 3 , &c., are the sums of the first, second, third, &c., powers of the roots of the equation. Multiplying out and equating coefficient. 6 !, we get (n - 2)g = S 2 (, l -3)r = S 3 &c. =&c. Or S S a +jpS 2 +gS 1 +3r=0 &c. &c. Ex. 1. As a particular case, take the cubic equation X s + qx + r = . Here S = The last may be written S 5 S 3 S 2 - nr - - 2 * 3 2 i.e., if a + b + c = 0, then will 532 From S 7 + ?S 5 + ?-S 4 = 0, we get Ex. 2. Take the biquadratic equation Here i.e., if a then S, 2 S, EQUATIONS WHOSE ROOTS ARE RATIONAL 113. It has been shown in Art. 79, that the last terra of any equation is always the product of its roots taken with contrary signs. Hence, when the roots are rational, they may be discovered by the following rule: Bring all the terms of the equation to one side; find all the divisors of the last term, and substitute them suc cessively for the unknown quantity. Then each divisor, which produces a result equal to 0, is a root of the equation.
Ex. Letx 3 -4#2
