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11. [Maximum mark: 23] This question asks you to investigate the roots of a polynomial. f(x) = ax3 + bx? + cx + d = 0 has roots a, B.Y. Let S,, = (" + B" + y". (a) Factorise ax3 + bx2 + cx + d, then expand your result with a. B, y. [3] (b) (i) Show that Sja + b = 0. (il) By expanding (a + 8 + y)?, or otherwise, show that Sza + S,b + 2c = 0. (ill) By considering a"-3f(a). "-3f(B). "-3f(), or otherwise, show that Sna + Sn-1b+ S,1-26+ Sn-3d=0. [7] Now you are given that x3 - x + d = 0, d = 0, has roots a, B. Y. (c) Find, in terms of d, the value of: (1) a+B+ Y (ii) a2+ 82 + 2 (ii1) 43+ 83 + 13 (iv) (v) 1 aB BY [9] (d) Find, in terms of d, an equation with roots a + B, Pty. a + y. [4]