As the article indicates, in order to properly understand partial fractions, one must be considerably familiar with irreducible polynomial:
For any field F, a polynomial with coefficients in F is said to be irreducible over F if it is non-constant and cannot be factored (other than a trivial factorization) into the product of two or more non-constant polynomials with coefficients in F.
This definition resembles that of prime numbers. Similarly, the factorization into irreducible polynomials is unique aside from the order of the factors.