I am trying to find all of the roots (real and imaginary) of this polynomial.

Question

I am trying to find all of the real and imaginary roots of this polynomial. $$y=9x^7-x^6-4x^5+2x^4-2x.$$

Answer 1

If we factor out an $x$ we get $x(9x^6-x^5-4x^4+2x^3-2)$. To find the zeroes, set resulting equation equal to $0$. One root is obviously $0$...the others would likely need to be found by numerical methods or by mathematical programs.

(I used Wolfram Alpha and found two other real roots and four imaginary roots - complex conjugate pairs.)

Answer 2

Dividing out the $x$, substituting $x\leftarrow x/9$, and clearing denominators, the equation to be solved transforms to $$x^6−x^5− 36 x^4+162 x^3−118098=0\text{.}$$ It turns out that the equation $$y^6 + 8y^3 + 32y^2 - 16y - 288=0$$ defines the same algebraic extension of the rationals, as witnessed by the pair of maps $$\begin{align}x&\leftarrow \frac{-y^5 - 8y^2 - 32y + 16}{16}\\ y&\leftarrow \frac{-x^5 + x^4 + 36x^3 - 162 x^2}{6561}\text{.}\end{align}$$ The Galois group of the latter equation can be calculated to be $S_6$. Therefore, this polynomial system is not solvable by radicals. What's more (as commenters have pointed out) the polynomial is irreducible.

Basically, algebra can't simplify this problem. By Sturm's theorem, apart from zero there are two real roots and four non-real roots—use a numerical root-extraction algorithm to find them.

I used Magma for these calculations:

R<x> := PolynomialRing(Rationals());
K<a> := NumberField(x^6-x^5-36*x^4+162*x^3-118098);
L<b>, M := OptimizedRepresentation(K);
f<y> := DefiningPolynomial(L);
f;
M(a);
N := Inverse(M);
N(b);
G, R, S := GaloisGroup(f);
G;

Output:

y^6 + 8*y^3 + 32*y^2 - 16*y - 288
1/16*(-b^5 - 8*b^2 - 32*b + 16)
1/6561*(-a^5 + a^4 + 36*a^3 - 162*a^2)
Symmetric group G acting on a set of cardinality 6
Order = 720 = 2^4 * 3^2 * 5