Let $t$ be the area of a regular pentagon with each side equal to $1$. Let $P(x)=0$ be the polynomial equation with least degree, having integer coefficients, satisfied by $x=t$ and the $gcd$ of all the coefficients equal to $1$. If $M$ is the sum of the absolute values of the coefficients of $P(x)$, What is the integer closest to $sqrt{M}$ ? ($sin 18^{circ}=(sqrt{5}-1)/2$)

Here in this problem I first started out by trying to find out the area of the pentagon, but that dosn’t really help. Also some people claimed that the value of $sin 18 neq (sqrt{5}-1)/2$ but instead it is $(sqrt{5}-1/4)$. That is another thing that bothers me, but I wanted to know about what is the motivation behind your solution.