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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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Are you familiar with Rational Roots Theorem? –  Rod Nov 18 '13 at 7:57

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up vote 12 down vote accepted

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.

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Oh ! that was neat... "Any rational root of a monic polynomial with integer coefficients is an integer" is obvious using the Rational Roots Theorem. Thanks ! –  Apurv Nov 18 '13 at 13:41

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