# 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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