MathLabs

Problem 4

A point in the plane with a Cartesian coordinate system is called a mixed point if one coordinate is rational and the other is irrational. Find all polynomials with real coefficients whose graphs contain no mixed point.
Step 4 of 7: Choose a large prime
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q=f(0)+p,p>max⁡{f(1)−f(0), all real roots of f(x)−f(0)−p}.q=f(0)+p,\qquad p>\max\{f(1)-f(0),\text{ all real roots of }f(x)-f(0)-p\}.
Detailed analysis

Choose a sufficiently large prime p. Then f(1)-q is negative, while f(p)-q is positive because f is monic of degree at least two and p is beyond all real roots of f(x)-q.