  In this paper, we develop a direct formula for determining the coefficients
in the canonical basis of the best polynomial of degree $M$ that approximates a
polynomial of degree $N>M$ on a symmetric interval for the
$\mathcal{L}^2$-norm. We also formally prove that using the formula is more
computationally efficient than using a classical matrix multiplication approach
and we provide an example to illustrate that it is more numerically stable than
the classical approach.
