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 : $\text{sgn}(x)=\begin{cases} 1 \quad , x>0 \\ 0 \quad , x=0 \\ -1 \quad , x<0 \end{cases}$.