Question 1171716
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If the polynomial *[tex \Large \rho(x)\ =\ 4x^2\,+\,bx\,-\,9] can be factored into integral first degree polynomials, then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{-b\,\pm\,\sqrt{b^2\,-\,(4)(4)(-9)}}{4}\,=\,\frac{-b\,\pm\,\sqrt{b^2\,+\,144}}{4}\,\in\ \mathbb{Q}]


For that to be true, the radicand must be a perfect square and then both the sum and difference of *[tex \Large -b] and the radical must be evenly divisible by 4.  Hence, the allowable values of *[tex \Large b] are:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \{b\,|\,\sqrt{ b^2 \,+\,144}\,\in\,\mathbb{Q}\,&\,\(-b\,\pm\,\sqrt{b^2\,+\,144}\)\,\text{mod} \,4 \,= \, 0\}]


The only number I can find that fits into that set is 16.

																
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
*[illustration darwinfish.jpg]

From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> 
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