Do you really wanna ask for even?

Geometry Level 3

f(x)=(a25a+4)x3(6{a}25{a}+1)x(tan(x))sgn(x)f(x)=\left( \lfloor a \rfloor^2-5\lfloor a \rfloor +4 \right)x^3-\left( 6\{a\}^2-5\{a\}+1 \right)x -(\tan (x)) \text{sgn}(x)

Provided that f(x)f(x) be an even function for all xRx \in \mathbb{R}. If sum of all possible values of aa is PQ\dfrac{P}{Q} for coprime positive integers P,QP,Q, then find the value of (P+Q)(P+Q).

Details And Assumptions:

  • ..\lfloor ..\rfloor is floor function or greatest integer function.

  • {..}\{ .. \} is fractional part function.

  • sgn(x)\text{sgn}(x) is signum function defined as : sgn(x)={1,x>00,x=01,x<0\text{sgn}(x)=\begin{cases} 1 \quad , x>0 \\ 0 \quad , x=0 \\ -1 \quad , x<0 \end{cases}.

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