Equation with high exponents

algebra-precalculus

I would appreciate any help with this problem: $ x^8+2x^7+2x^6+5x^5+3x^4+5x^3+2x^2+2x^1+1x^0=0 $

I know that when $x$ isn't zero $x^0=1$ so the equation could be re-written as $ x^8+2x^7+2x^6+5x^5+3x^4+5x^3+2x^2+2x+1=0 $. I am not sure what to do from here. I have tried using wolframalpha to get the solutions so I know real ones exist (2 of them, actually), but I have no idea how to get them.


As the equation is Reciprocal Equation of the First type,

Divide either sides by $x^4,$ to get $$x^4+\frac1{x^4}+2\left(x^3+\frac1{x^3}\right)+2\left(x^2+\frac1{x^2}\right)+5\left(x+\frac1x\right)+3=0$$

Put $x+\frac1x=y$ to reduce the equation the degree $\frac82=4$

Can you take it form here?

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