Suppose that P = (33,31) is a point on the cubic curve '92 = m3+ax2+bcc+c (a) Verify that the x-coordinate of the point 2P is given by the...

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1.19. Suppose that P = (33,31) is a point on the cubic curve’92 = m3+ax2+bcc+c(a) Verify that the x-coordinate of the point 2P is given by the duplication fommla m4 — 213322 — 8cm + b2 — 4cm4x3 + 4ax2 + 413x + 4c(b) Derive a similar formula for the y-coordinate of 2P in terms of ac and y. (c) Find a polynomial in 9: whose roots are the x-coordinates of the point P =($41) satisfying 3P = O. (Him. The relation 3P = O can also be written as2P = —P.) (d) For the particular curve 3/2 = x3 + 1, solve the equation in (c) to ﬁnd all pointssatisfying 3P = 0. Note that you will need to use complex numbers. 93(2P) =