SetDirectory[NotebookDirectory[]]

Helper methods

<< Bezout.m

x[coef_, var_, exp_] := Plus @@ Table[(Subscript[coef, j])(var^j), {j, 0, exp}]

BezoutDet = (Subscript[v, i_,j_]) :>xs[[i+1]] ys[[j+1]] - xs[[j+1]] ys[[i +1]]

Define some parametric curves

x1 = x[cx1, t, 3]

x2 = x[cx2, s, 3]

y1 = x[cy1, t, 3]

y2 = x[cy2, s, 3]

Set axes equal to one another, grouped by ‘t’

xs = CoefficientList[x2 - x1, t]

ys = CoefficientList[y2 - y1, t]

Find Bezout of ‘xs’ and ‘ys’

Make sure the format of the Bezout is as expected

BezoutMatrix[3] //MatrixForm

View resulting Bezout in various forms

Collect[Expand[BezoutMatrix[3] /. BezoutDet], s] // MatrixForm

Simplify[BezoutMatrix[3] /. BezoutDet] //MatrixForm

Rewrite matrix elements as generic polynomials

Each of these coefficients is calculated in code using the members of the Bezout above

m = {{a, b, c0 + s c1+(s^2)c2+(s^3)c3},{d ,e0 + s e1+(s^2)e2+(s^3)e3, f0 + s f1+(s^2)f2+(s^3)f3},{g0 + s g1+(s^2)g2+(s^3)g3, h0 + s h1+(s^2)h2+(s^3)h3, i0 + s i1+(s^2)i2+(s^3)i3}}

MatrixForm[m]

Find determinant of Bezout matrix

The determinant is then calculated from the above coefficients

CoefficientList[Collect[Expand[Det[m]],s],s] // TableForm