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 6 of 7: Use the rational-root theorem
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f(x)−q∈Z[x] monic,q−f(0)=p  ⟹  r∈{1,p,−1,−p}∩(1,p),f(x)-q\in\mathbb Z[x]\text{ monic},\quad q-f(0)=p\implies r\in\{1,p,-1,-p\}\cap(1,p),
Detailed analysis

Any rational root of the monic integer polynomial f(x)-q is an integer dividing its constant term, whose absolute value is p. The only possible positive rational roots are 1 and p, but r is strictly between them; hence r is irrational, contradicting the graph condition at the rational ordinate q.