$\pi$ polynomials whose real zeros approximate $\pi$ Let's have the following polynomials
$$x^4+105x^2-1134=0,$$
$$x^6+126x^4+10395x^2-115830=0,$$
$$3x^8+550x^6+45045x^4+3378375x^2-38288250=0$$
The positive real zeros of these equations are good approximations of $\pi$. Does anyone know how to formulate the next polynomial so its real positive zeros give a better aproxmation of $\pi$?
 A: How do you measure the quality of these approximations?  Here are the errors in the roots of your polynomials:
$$
\begin{eqnarray}
\text{deg}&=&4; \qquad \rho=3.1419530007425911; \qquad\epsilon=3.6\times10^{-4}\\
\text{deg}&=&6;\qquad \rho=3.1415990271727633; \qquad \epsilon=6.4\times10^{-6}\\
\text{deg}&=&8; \qquad\rho=3.1415927638681944; \qquad \epsilon=1.1\times10^{-7}.
\end{eqnarray}
$$
I would argue that these are fairly inefficient... you've used $24$ digits of coefficients to get only $7$ correct digits of $\pi$, for instance, in the degree-$8$ example.  It would be more efficient to just use $1000000000x - 3141592653=0$!
There is a terrific compilation of approximations for $\pi$, including polynomial-root approximations, over at The Contest Center's Pi Competition.  My favorite is $$6x^6-4x^5+5x^4+2x^3-2x^2+3x-5083=0,$$
which has just ten digits of coefficients and leads to the fourteen-digit approximation
$$
\rho=3.1415926535898031685143792;\qquad\epsilon=1.0\times10^{-14}.
$$
The best approximation of any kind on the page is
$$
\frac{\log\left((5!\times5336)^3 + 4! + 6!\right)}{\sqrt{163}},
$$
which is correct to thirty decimal places.
A: From ancient relation, $\frac{\pi}{(\phi+1)}= \frac{6}{5}$ I had such approximation
(not very good, but ancient)
$25x^2 - 90x + 36$
error $3\times10^{-3}$
