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Even degree and Odd coefficient Crossed Polynomial Poser (Posted on 2023-10-29) Difficulty: 3 of 5
P(x) is a polynomial of even degree. Also, all the coefficients of P(x) are odd numbers.

Is it possible for P(x) to have a rational root?
• If so, provide an example.
• If not, prove that it is NOT possible for P(x) to have a rational root.

No Solution Yet Submitted by K Sengupta    
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re: proof for quadratics Comment 7 of 7 |
(In reply to proof for quadratics by xdog)

I was a little sceptical, but I see that xdog is right.  


Odd squares = 1 mod 8
Even squares = 0 or 4 mod 8

If a and c are odd integers, 4ac = 4 mod 8.

b^2 - 4ac = 5 mod 8, so D is not a an odd or an even square, so the equation has no rational roots.



  Posted by Steve Herman on 2023-10-30 20:56:01
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