If $ \alpha, \beta ,\gamma , \delta $ are the roots of $x^4+x^2+1=0$ then the equation whose roots are $\alpha^2 , \beta^2, \gamma^2, \delta^2$ is

Let y = x^2, which allows the original equation x^4 + x^2 + 1 = 0 to be rewritten as y^2 + y + 1 = 0. The roots of this equation in y are the squares of the roots of the original equation, meaning y takes the values alpha squared, beta squared, gamma squared, and delta squared. To find the equation whose roots are these squared values, we replace y with x, yielding x^2 + x + 1 = 0. Since this is a quadratic equation and the original was a quartic, the required equation with all four corresponding roots is the square of this quadratic, which is (x^2 + x + 1)^2 = 0.

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