In the following text about Gaussian quadrature by Brian Bradie I cannot understand the meaning of the author:
Associated with each weight function is a special family of polynomials, unique up to a normalization factor.
What is the meaning of the unique up to a normalization factor? There are books with the same statement (i.e. unique up to a normalization factor) in other topics.
Usually it means "unique up to a re-scaling of the independent variable." For example, $x^2+x+1$ is essentially the same thing as $4x^2+2x+1$ under the transformation $x \mapsto 2x$.
The idea is that we can arbitrarily re-scale our domain to make life more convenient, for instance changing a length scale from meters to kilometers, or choosing a scaling of an orthogonal polynomial family such that $\langle P_i P_j \rangle = \delta_{ij}$.