Do you really wanna ask for even?

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

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

Details And Assumptions:

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

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

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