# A solvable nonic polynomial

Continuing from our demonstration that a certain sextic polynomial, which are not in general solvable, has an explicit factorization, we go on to describe how a class of degree 9 polynomials is solvable. Consider $a,b,c,d$ to be rational integers, not all zero, and the nonic polynomial

$F(x) = x^9 + a x^8 + b x^7 + c x^6 + d x^5 - (126 - 56 a + 21 b - 6 c + d)x^4$

$- (84 - 28 a + 7 b - c)x^3 - (36 - 8a + c)x^2 - (9 - a)x - 1.$

The claim is that all such polynomials are in fact solvable!

I will reveal the argument a little later, but it’ll be interesting to see what kind of arguments readers can come up with.