Do the root shuffle

Algebra Level 5

\(f(x)\) and \(g(x) \) are monic quadratic polynomials that satisfy the following conditions:

  1. \(f(x) = 0 \) has real distinct roots \( a_1 \) and \(a_2\).
  2. \(g(x)= 0 \) has real distinct roots \( b_1\) and \( b_2\).
  3. \( \{ f(b_1), f( b_2) \} = \{ b_1, b_2 \} \).
  4. \( \{ g( a_1), g(a_2) \} = \{ a_1, a_2 \} \).
  5. \(f(1)g(1) = 132 \).

What is the value of \(f(2)g(2)?\)

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial \( x^3 + 3x - 5 \) is monic but the polynomial \( -x^4 + 2x^3 - 6 \) is not.

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