Polynomial Synthetic Division [A fifth Example:]

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Here is an example on dividing by a factorable quadratic through repeated application of Synthetic Division [in other words, going around the handicap of Synthetic Division.]

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Take requests? I don't get partial derivatives. Do (x+i*y)^n or some of these. My education is grade 12 pre-cal completed. I don't have anything after that.

Ellipse Positive Set, x^2+x*y+y^2=1, <(1/3)*sqrt(3)*cos(ϑ)-sin(ϑ),
[x/sqrt(x^2+x*y+y^2), y/sqrt(x^2+x*y+y^2)]
[(x^2-y^2)/(x^2+x*y+y^2), y*(y+2*x)/(x^2+x*y+y^2)]
[(x^3-3*x*y^2-y^3)/(x^2+x*y+y^2)^(3/2),
[(x^4-6*x^2*y^2-4*x*y^3)/(x^2+x*y+y^2)^2,
[(x^5-10*x^3*y^2-10*x^2*y^3+y^5)/(x^2+x*y+y^2)^(5/2),

Ellipse Negative Set, x^2-x*y+y^2=1, <cos(ϑ)-(1/3)*sqrt(3)*sin(ϑ),
[x/sqrt(x^2-x*y+y^2), y/sqrt(x^2-x*y+y^2)]
[(x^2-y^2)/(x^2-x*y+y^2), y*(-y+2*x)/(x^2-x*y+y^2)]
[(x^3-3*x*y^2+y^3)/(x^2-x*y+y^2)^(3/2),
[(x^4-6*x^2*y^2+4*x*y^3)/(x^2-x*y+y^2)^2,
[(x^5-10*x^3*y^2+10*x^2*y^3-y^5)/(x^2-x*y+y^2)^(5/2),

thomasolson