Consider a cubic function with real coefficients and complex roots. Later must be of the form a+ib, a-ib (complex conjugate), hence the form of the function: c*(a+ib-x)*(a-ib-x)*(x-r) = c*((a-x)2+b2)*(x-r), r being a real root. The derivative being c*((a-x)2+b2)-c*2*(a-x)*(x-r), its value at x=a is c*b2. The tangent at this point is easily calculated and has the equation y = c*b2*(x-r), thus passing through the real root of the cubic.
Some further properties are discussed below. For conveniance we assume c=1. It is easy to find the second point for which the tangent is parallel to the previous one. This happens for x = e = (a+2*r)/3. The two corresponding points A, B on the graph have parallel tangents and the middle M of the line AB coincides with the symmetry center of the cubic, thus locates its inflexion point, where the second derivative is zero, at g = (e+a)/2 = (2*a+r)/3. Let C be the other intersection point of the qubic with the tangent at B. (r,0) and the projections of the four points A, B, M and C on the x-axis are equidistant, the common distance being (a-r)/3.
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Cubic with complex roots ![[0_0]](CubicWithComplexRoots_files/CubicWithComplexRoots_0_0_0.png)
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