Make sure you read part 1 first! Recall that we were exploring integrating rational functions, and to do so, we needed to look at partial fraction decompositions. As we now begin to discuss that in detail, we first look more closely at polynomial rings (over fields).
Polynomial rings over fields
Polynomial division with remainder, and the factor theorem
Evaluating polynomials in (extension) fields
Conjugate root theorem for real polynomials
Complete factorisation of real polynomials
Theorem. Every real polynomial \(p \in \mathbb R[\mathrm x]\) can be factorised over \(\mathbb R\) into a product of linear and irreducible quadratic factors
Existence and uniqueness of partial fraction decompositions
It also works over any Euclidean domain!!
Partial fractions for the rationals: an algorithm
Solving the original problem: integrating rational functions
To be continued…