x^6+x^5-2x^4-x^3+x^2+1=0

Y=x^4-x^2+x-1

show y satisfies a quadratic equation with rational coefficients.

I've tried and have not be able to do so, so far.

Do it, please.

Thanks.

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- August 29th 2010, 10:47 AMberachiaTripos question 1903
x^6+x^5-2x^4-x^3+x^2+1=0

Y=x^4-x^2+x-1

show y satisfies a quadratic equation with rational coefficients.

I've tried and have not be able to do so, so far.

Do it, please.

Thanks. - August 29th 2010, 11:53 AMOpalg
I think this is just an exercise in division of polynomials. Square y, then divide by the expression that is given to be zero:

. . . . .

Therefore y satisfies the quadratic equation .

(I'll leave you to convince yourself that this is an occasion where division by zero is justified!)