What is the Value of (p+q-2t) and (p^2+q^2-2t^2) in an Equation with Degree 3?

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The discussion focuses on finding the values of (p+q-2t) and (p^2+q^2-2t^2) for the roots p, q, and t of the cubic equation 8x^3+4cx^2+2(b-3)x+(a-2c)=0. The user is guided to express the polynomial in factored form and compare coefficients to derive relationships between the roots and the coefficients. Key relationships provided include p+q+t=c/2, pq+pt+qt=(b-3)/4, and pqt=(2c-a)/8. However, the user specifically seeks assistance in calculating the values of (p+q-2t) and (p^2+q^2-2t^2). The thread emphasizes the need for further clarification on these specific expressions.
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I have an equation similar:
8x^3+4cx^2+2(b-3)x+(a-2c)=0
a, b, c are real number.
if p, q , t be roots.
What can say about (p+q-2t) and (p^2+ q^2-2t^2)?
Please help me??
Thanks so much.
 
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hint: write your polynomial as 8(x-p)(x-q)(x-t) and multiply it out, then compare coefficients.
 
I know
p+q+t=c/2
pq+pt+qt=(b-3)/4
pqt=(2c-a)/8
but I need value for
(p+q-2t)??
(p^2+ q^2-2t^2)?
Please help me??
 
I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...

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