Classical integrable finite-dimensional systems related to by Olshanetsky M.A., Perelomov A.M.

By Olshanetsky M.A., Perelomov A.M.

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Assume the result is true for n = k, so that t; When n = k = + 1, (1 - 12)(1 - ;2) .. '(1- ~2) = k ~ 1. 3) the LHS is T k + 1 , where Tk + 1 T k [ 1 - (1 : kf = J. 3) , we may write this as Tk + 1 = k + 1[1 _ 1 ] = (k + 1)[(k + 1)2 - 1] 2k (k 2 + 2k) 2k(k + 1) (1 + k) 2 (k + 2) 2(k + 1) 2k(1 (k + 1) 2(k + + + k) 2 1 1) . 3) with k replaced by (k + 1), and so the result is true when n = k + 1. Hence, by induction, the result is true for all integers n ;;;,: 2. Example 16 = 5 and The sequence of numbers Ut.

7 The polynomial f(x) == x 3 + ax 2 + bx + e leaves remainders 7,1 ,19 on division by (x - 1), (x + 1), (x - 2), respectively. Find a, b , e and the remainder when f(x) is divided by (x + 2) . 8 Show that (x + 3) is a factor of x 4 + 2x3 + 7x 2 + llx - 57. 9 Find the values of a and b if (x - 1) and (x + 2) are both factors of ax 3 + 3x 2 + bx - 2, and state the third factor. 10 (a) Show that (x - e) is a factor of x 3 - e3 and hence show that x3 - (b) Show that (x + e) is a factor of x + 3 x 3 + e3 == (x- e)(x 2 + 2 == (x + e)(x 2 2 ex + 2) .

A = 2. Substituting x = -2 =? B = 3. Hence , the given function is equal to 1 + 2 (x + 1) + 3 (x + 2) . Polynomials and rational functions 27 Exercise 2 1 Add together 3x3 + 2x + 6, 2x2 + X + 1 and 2x3 + x 2 + 3. 2 Subtract 5x3 + 2x2 - 3x + 1 from 6x3 + 8x + 5. 3 Multiply (x 3 + 2x + 1) by (x 2 + X + 1) (a) by multiplying the brackets, (b) by long multipl ication . 4 Divide x 3 + 5x2 + llx + 10 by (x + 2). 5 Find the quotient and the remainder when x 3 + 3x 2 + 2x is divided by x 2 - X + 1. 6 Find the remainder when (a) 5x3 + 2x2 + X + 1 is divided by (x + 1), (b) 6x4 + 2x3 + 3x2 + X + 1 is divided by (2x - 1).

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