Derived Polynomial. 172. Corollary. If two or more of the roots of the given equation are equal to each other, such as x = x = x = &c., the roots of the preceding equation become x - x = 0, x - x = 0, x - x, x - x, &c. and it is, therefore, satisfied by the value of u, which reduces it to u = 0, nxin-1 + (n-1) ax'n-2 +(n-2) bx/n-3+ &c. = 0, or since x' is either of the equal roots the accent may be omitted, and we have nxn-1+(n-1) axn-2 +(n-2) bxn-3+ &c. = 0, which must be satisfied by a value of x equal to either of the equal roots of the given equation, and these equal roots can therefore be obtained by means of the process of elimination of art. 116. But it is evident that two different equations with one unknown quantity cannot be satisfied by the same value of this unknown quantity, unless their first members have a common divisor, which is reduced to zero by this value of the unknown quantity. The first member of the equation last obtained is called the derived polynomial of the given equation, and is obtained from it by multiplying each term by the exponent of the unknown quantity in that term, and diminishing this exponent by unity. The equal roots of an equation are, therefore, obtained by finding the greatest-common divisor of its first member and its derived polynomial, and solving the equation obtained from putting this common divisor equal to zero. the greatest common divisor of which and the given first member is Now since the given equation has two roots equal to 2, it must be divisible by and we have x3 whence (x - 2)2 = x2 - 4 x + 4, 7 x2 + 16 x - 12 = (x - 2)2 (x-3) = 0; is the other root of the given equation. 2. Find all the roots of the equation x7-9x5+6x4 + 15x3 - 12x2-7x+6=0 which has equal roots. Solution. The derived polynomial of this equation is 7x6-45x4+24x3 +45x2-24х-7, the greatest common divisor of which and the given equation gives Examples of finding equal Roots. which is an equation of the third degree, and we may consider it as a new equation, the equal roots of which are to be found, if it has any. and the common divisor of this derived polynomial and the first member gives x3 -x2-x+1=(x-1)2 (x + 1) = 0. The equal roots of the given equation are, therefore, X = 1, and = -1; and its first member is divisible by (x - 1)3 (x + 1)2, and is found by division to be (x - 1)3 (x + 1)2 (x2 + x - 5). The remaining roots are, therefore, found from solving the 3. Find all the roots of the equation x3+3x2-9x-9=0 8. Find all the roots of the equation x2 + 12x3 + 54x2 + 108 x + 81 = 0 which has equal roots. 10. Find all the roots of the equation x6-6x4+4x3 + 9x2 - 12x+4=0 Examples of finding equal Roots. 11. Find the equal roots of the equation x8-8x7+26x6-45x5+45x4-21x310x2+20x-8=0. Ans. x = 1, or = 2. 172. General solutions have been given of equations of the third and fourth degree; they are not, however, always applicable, and are so complicated, that it is more convenient in practice to obtain the roots of an equation by successive approximations. But before proceeding to this subject, it is important to be acquainted with the theory of arithmetical and geometrical progressions. |