Let c be a nonzero real number. Suppose that g(x)=c0xr+c1xr−1+⋯+cr−1x+cr is a polynomial with integer coefficients. Suppose that the roots of g(x) are b1,⋯,br. Let k be a given positive integer. Show that there is a prime p such that p>max(k,∣c∣,∣cr∣), and moreover if t is a real number between 0 and 1, and j is one of 1,⋯,r, then ∣(crbjg(tbj))pe(1−t)b∣<2r(p−1)!. Furthermore, if f(x)=(p−1)!erp−1xp−1(g(x))p then j=1∑r∫01e(1−t)bjf(tbj)dt≤21.