Consider the problem of fitting a cubic through m given points P1(x1, y1), . . . , Pm(xm, ym) in the plane. A cubic is a curve in ℝ 2 that can be described by an equation of the form f (x, y) = c1 + c2x + c3 y + c4x2 + c5xy + c6 y2 + c7x3 + c8x2 y + c9xy2 + c10 y3 = 0, where at least one of the coefficients ci is nonzero. If k is any nonzero constant, then the equations f (x, y) = 0 and k f (x, y) = 0 define the same cubic.
Show that the cubics through the points (0,0), (1,0), (2,0), (0,1), and (0,2) can be described by equations of the form c5xy +c7(x3 −3x2 +2x)+c8x2 y +c9xy2 + c10(y3 − 3y2 + 2y) = 0, where at least one of the coefficients ci is nonzero. Alternatively, this equation can be written as c7x(x −1)(x −2)+c10 y(y −1)(y − 2) + xy(c5 + c8x + c9 y) = 0.
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