Do the root shuffle

Algebra Level 5

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

  1. f(x)=0f(x) = 0 has real distinct roots a1 a_1 and a2a_2.
  2. g(x)=0g(x)= 0 has real distinct roots b1 b_1 and b2 b_2.
  3. {f(b1),f(b2)}={b1,b2} \{ f(b_1), f( b_2) \} = \{ b_1, b_2 \} .
  4. {g(a1),g(a2)}={a1,a2} \{ g( a_1), g(a_2) \} = \{ a_1, a_2 \} .
  5. f(1)g(1)=132f(1)g(1) = 132 .

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

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x5 x^3 + 3x - 5 is monic but the polynomial x4+2x36 -x^4 + 2x^3 - 6 is not.

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