Example $(x-1)(x-8)(x-31)-1$. Just by looking at this polynomial how do you conclude that the roots are irrational?
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Any rational root of a monic polynomial with integer coefficients is an integer. And the product $(x-1)(x-8)(x-31)$ of three integers can equal $1$ only if all three terms are $\pm 1$. This is clearly not possible, so there are no rational roots.