Roots of a polynomial of degree 4

[Mathman23]

(*)p(x) = x^4 + ax^3 + bx^ 2 + ax + 1 = 0

where a,b \in \mathbb{C}

I would like to prove that a complex number x makes (*) true iff

s = x + x^{-1} is a root of the Q(s) = s^2 + as + (b-2)

I see that that Q(x + x^{-1}) = \frac{p(x)}{x^2}

Then to prove the above do I then show that p(x) and Q((x + ^{-1}) shares roots?

Sincerely Yours
MM23

 

[arildno]

Yes. Note that Q(s) and p(x) must have coincident roots.

 

[HallsofIvy]

(*)p(x) = x^4 + ax^3 + bx^ 2 + ax + 1 = 0

where a,b \in \mathbb{C}

I would like to prove that a complex number x makes (*) true iff

s = x + x^{-1} is a root of the Q(s) = s^2 + as + (b-2)

I see that that Q(x + x^{-1}) = \frac{p(x)}{x^2}

Then to prove the above do I then show that p(x) and Q((x + ^{-1}) shares roots?

Sincerely Yours
MM23

The onlly way a fraction can be 0 is if the numerator is 0. Isn't it obvious from Q(x + x^{-1}) = \frac{p(x)}{x^2}
that Q(x+ x^-1)= 0 if and only if p(x)= 0?