If $\displaystyle \lambda, \mu, v, \lambda', \mu', v' $ are rational numbers find rationals numbers $\displaystyle \lambda'', \mu'', v'' $such that:

$\displaystyle (\lambda\alpha^2 + \mu\alpha + v)(\lambda'\alpha^2 + \mu'\alpha + v') = \lambda''\alpha^2 + \mu''\alpha + v'' $

$\displaystyle (\lambda\alpha^2 + \mu\alpha + v)(\lambda'\alpha^2 + \mu'\alpha + v') = \lambda \lambda'\alpha^4 + (\mu\lambda' + \lambda\mu')\alpha^3 + (v\lambda' + \mu\mu' + \lambda v')\alpha^2 $ $\displaystyle + (v \mu' + \mu v')\alpha + vv' $

Using the result in the previous posts $\displaystyle \alpha^4 = \alpha^2 + \alpha $

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So we put:

$\displaystyle \lambda'' = \lambda\lambda' + \mu\mu' + \lambda v $

$\displaystyle \mu'' = \lambda\lambda' + \mu\lambda' + \lambda\mu' +v\mu' + \mu v' $

$\displaystyle v'' = \mu\lambda' + \lambda\mu' + vv' $

Is this the correct method for this question...I'm not sure since the result I used was in {0,1} whereas the question did not specify this...?