# Integral of absolute value of polynomial?

Let $a(x)$ and $b(x)$ be integer irreducible polynomials where $b$ is U-shaped in the interval mentioned below and has 2 distinct real zeros. The zeros of $b$ cannot be expressed by radicals. Also $b$ is the derivative of $a$ and $b$ has degree prime of type 7 mod 8.

Is it possible that there exists a nonzero integer $m$ such that the interval $[-m,m]$ contains the two zeros of $b$ and the integral from $-m$ to $m$ of the absolute value of $b(x)$ $dx$ can be given in rootform?

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What's rootform? – joriki Sep 20 '12 at 22:04
@joriki : wild guess would be "expressible by radicals"? But it deserves clarification... – Patrick Da Silva Sep 20 '12 at 22:22
If $b$ has odd degree, its graph can't be U-shaped. – Gerry Myerson Sep 21 '12 at 0:20
Ah sorry , i mean U shaped in the interval. – mick Sep 21 '12 at 13:41
If $\alpha,\beta$ are the two roots of $b$, then $$\int_{-m}^m|b(x)|\,\mathrm dx = \int_{-m}^mb(x)\,\mathrm dx-2\int_{\alpha}^\beta b(x)\,\mathrm dx=a(m)-a(-m)-2a(\beta)+2a(\alpha)$$ So the question is whether $a(\alpha)-a(\beta)$ can be expressed by radicals. – Hagen von Eitzen Apr 25 '15 at 11:47