Theory of equations

Question

If f(x) is a polynomial of degree 4 with rational coefficients and touches x-axis at $(\sqrt{2},0)$, then for the equation f(x) = 0,

Moderate

Solution

It is given that f(x) touches x-axis at $(\sqrt{2},0)$. So, two roots are $\sqrt{2},\sqrt{2}$.

Sine coefficients are rational, other roots must be $-\sqrt{2},-\sqrt{2}$.

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