x^2=X
ax+b=0
a^2X=b^2

x^3=X
ax^2+bx+c=0
a^3X^2+b^3X+c^3=3abcX

x^4=X
ax^3+bx^2+cx+d=0
X^3a^4+X^2(-2a^2c^2-4a^2bd-b^4+4ab^2c)+X(c^4+2b^2d^2-4bc^2d+4acd^2)=d^4

x^5=X
ax^4+bx^3+cx^2+dx+f=0
X^4a^5
+X^3(b^5-5ab^3c+5a^2b^2d+5a^2b(-af+c^2)-5a^3cd)
+X^2(c^5+5c^3(-bd-af)+5c^2(ad^2+b^2f)+5c(-abdf+b^2d^2+a^2f^2)+5ab^2f^2-5b^3df+5a^2d^2f-5abd^3)
+X(d^5-5cd^3f+5bd^2f^2+5df^2(c^2-af)-5bcf^3)
+f^5
=0

x^7=X
ax^6+bx^5+cx^4+dx^3+fx^2+gx+h=0
X^6a^7+X^5(7a^4b^2g-7a^3b^3f-21a^3b^2cd+7a^2b^4d+14a^2b^3c^2-7ab^5c+b^7-7a^5bh+7a^4bd^2+14a^4bcf-7a^3bc^3-7a^5df-7a^5cg+7a^4c^2d)
+X^4(14a^3c^2g^2+14a^2c^3f^2-21a^2c^3dg+35a^2bc^2dh+7a^2c^4h+7ac^4d^2-7ac^5f-21abc^2d^3+14abc^4g-21ab^2c^3h+7b^4c^2h-7b^3c^3g-21b^3c^2df+7b^2c^4f+14b^2c^3d^2-7bc^5d+c^7+7a^4ch^2-7a^3cf^3+7a^3cdfg+7a^3bcgh-14a^2bcdf^2-7a^2b^2cg^2-14a^2b^2cfh+7ab^3cfg+7ab^3cdh+35ab^2cd^2f+7b^4cf^2+14b^4cdg-7b^3cd^3-21a^3c^2fh-21a^3cd^2h+7a^2cd^4-7a^2c^2d^2f-14a^2bc^2fg+35a^2bcd^2g-14ab^2c^2dg-7ab^2c^2f^2+7abc^3df+7a^4fg^2+14a^4dgh+7a^4f^2h+14a^3d^2f^2-21a^3bdg^2-21a^3bf^2g+14a^3b^2h^2+7a^3bdfh-21a^2b^3gh+35a^2b^2dfg+14a^2b^2f^3-21a^2bd^3f-7a^2b^2d^2h+7ab^4g^2+14ab^4fh-21ab^3d^2g-21ab^3df^2-7a^3d^3g+7ab^2d^4-7b^5fg-7b^5dh+7b^4d^2f)
+X^3(14a^2d^2g^3-14a^2d^2fgh+14a^2d^3h^2-7ad^3f^3-21acd^3g^2+14ad^4fg-14abcd^2h^2+35abd^2f^2h+35ac^2d^2gh-7ad^5h+14b^3d^2h^2+14b^2d^3g^2-21b^2d^3fh+7bd^4f^2-7bd^5g-21bcd^2f^3+14bcd^4h-7c^3d^3h-21c^3d^2fg+7c^2d^4g+14c^2d^3f^2-7cd^5f+d^7-7a^3dh^3-14a^2bdgh^2+35a^2cdfh^2-14a^2cdg^2h-21a^2df^3h+35ab^2dg^2h-14ab^2dfh^2-21ac^3dh^2-14ac^2df^2h+7abcdg^3+7abdf^3g-7b^3dg^3+7b^3dfgh-14b^2cdfg^2-7b^2c^2dh^2+7bc^3dgh+35bc^2df^2g-7a^2df^2g^2+14acdf^4+7c^4dg^2+14c^4dfh-7c^3df^3+7b^2df^4-7a^3g^3h-21a^3fgh^2+35a^2bfg^2h+35a^2cf^2gh-21a^2bch^3-7a^2bf^2h^2-7a^2c^2gh^2-7ab^3h^3-14ab^2f^2gh+35ab^2cgh^2+7abd^3gh-105abcdfgh-14abc^2g^2h+7abcf^3h+35abc^2fh^2+7ac^3fgh+7acd^3fh+7a^2bg^4-21ab^2fg^3+35abcf^2g^2-14abd^2fg^2+7b^4gh^2+14b^3f^2g^2-21b^3cg^2h-7b^2d^2f^2g+14b^2c^2g^3-21b^2cf^3g+35b^2c^2fgh-14b^2cd^2gh-21bc^3fg^2-7bc^2d^2g^2+7bcd^3fg-7b^3f^3h-21b^3cfh^2+35b^2cdf^2h+7bc^4h^2-21bc^3f^2h-14bc^2d^2fh+7a^2f^4g-21a^2cfg^3-7ac^3g^3+35ac^2dfg^2-21ac^2f^3g-14acd^2f^2g+7bc^2f^4+7c^4f^2g-7c^5gh-7abf^5)
+X^2(7af^2g^4+35adf^2gh^2+7af^4h^2-21af^3g^2h+14b^2f^2h^3-7bf^3g^3-21bdf^3h^2+14bf^4gh+14c^2f^3h^2+7cf^4g^2-7cf^5h-21cdf^2g^3+f^7-7df^5g+7d^2f^4h+14d^2f^3g^2-21d^3f^2gh+7a^2fh^4+7adfg^3h+7abfgh^3-14acfg^2h^2-7b^2fg^2h^2+7bcfg^3h+7bcdfh^3-7c^3fh^3-14c^2dfgh^2+35cd^2fg^2h+14bdfg^4+7c^2fg^4-7d^3fg^3-21acf^2h^3-21ad^2fh^3+7d^4fh^2+14acg^4h+7ac^2h^4+7acdgh^3+7b^2ch^4-21bc^2gh^3+35bcdg^2h^2-14bcf^2gh^2+14c^3g^2h^2-7c^2f^2g^2h+14c^2d^2h^3-21c^2dg^3h+7cdf^3gh-7cd^2f^2h^2-21cd^3gh^2+14a^2g^2h^3+14abdh^4-21abg^3h^2-7ad^2g^2h^2+7b^2g^4h-21b^2dgh^3-14bdf^2g^2h+35bd^2fgh^2-21bd^2g^3h-7bd^3h^3-7adg^5+7cd^2g^4+7d^4g^2h-7bcg^5)
+X(7bg^2h^4-7cg^3h^3+g^7-7fg^5h+7dg^4h^2+14f^2g^3h^2-21dfg^2h^3-7agh^5+14cfgh^4-7f^3gh^3+7d^2gh^4-7bfh^5+7df^2h^4-7cdh^5)
+h^7
=0