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December 13th, 2015, 01:52 AM   #1
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Quadratic with odd coefficients

Have no clue where to start a thread - but I ended up here where I can actually write something so i'm gonna just post my question I need help with! Thanks and please respond.

Suppose that a quadratic P(x) has all odd coefficients. Prove that P(x) = 0 has no rational roots.

Polynomial: is not given. Above is the exact given question.
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December 13th, 2015, 02:41 AM   #2
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The quadratic can be written as (ax + p)(bx + q) = abx² + (aq + bp)x + pq, where a, b, p and q are integers, if there are rational roots. If the coefficients ab and pq are odd, a, b, p and q must all be odd. It follows that aq and bp are odd, and so the coefficient aq + bp is even. Hence there cannot be rational roots if all the coefficients are odd.
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