Question

Let P₁(x) = x² +a₁x + b₁ and P₂(x) = x² +a₂x + b₂ be two quadratic polynomials with integer coefficients. Suppose a₁ ≠ a₂ and there exist integers m ≠ n such that P₁(m) = P₂(n), P₂(m) = P₁(n). Prove that a₁ – a₂ is even.

Answer 1

P₁(x) = x² +a₁x + b₁

P₂(x) = x² +a₂x + b₂         a₁,b₁,a₂,b₂ are integers

Now, as m, n are integers so (m + n) is also an integer and 

∴ a₁ + a₂ must be even integer. It is possible only when both a₁ and a₂ are even or odd. In both the cases

we get (a₁ – a₂) always be even.