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 2 of 7: Show that the coefficients are rational
Detailed analysis
If the degree is d, choose d+1 distinct rational inputs. Lagrange interpolation expresses f as the unique polynomial through those values; all interpolation data are rational, so every coefficient is rational.