Reduce the degree and solve the polynomial equation $x^6+ax^4-2x^3+1=0$ by algebraic tricks

Question

Reduce the degree of the polynomial and solve by algebraic tricks: $$x^6+ax^4-2x^3+1=0$$ where $a\in\mathbb R$.

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$a=0$ is obviously trivial. I tried all possible algebraic variations.

$$\frac {P(x)}{x^n}$$ where $P(x)=x^6+ax^4-2x^3+1$. For example,

$$\frac {P(x)}{x^3}=x^3+ax-2+\frac 1{x^3}=x^3+\frac 1{x^3}+ax-2=0$$

None of these tricks worked, because I couldn't spot the palindromic property. Do you think it would make sense to apply the factorization method?

Answer 1

Letting $c\in\mathbb{C}$ be such that $c^2=-a$, we get \begin{align*} &\, x^6+ax^4-2x^3+1 \\[4pt] =&\, (x^6-2x^3+1)-c^2x^4 \\[4pt] =&\, (x^3-1)^2-(cx^2)^2 \\[4pt] =&\, \bigl((x^3-1)+cx^2\bigr) \bigl((x^3-1)-cx^2\bigr) \\[4pt] =&\, (x^3+cx^2-1) (x^3-cx^2-1) \\[4pt] \end{align*} so solving the given equation reduces to solving two cubic equations.