9.6 Polynomials tutorial
Arithmetic properties of \(R[x]\):
(S1) \(\left[ p(x) + q(x) \right] + r(x) = p(x) + \left[ q(x) + r(x) \right] \quad \text{for all} \quad p(x), q(x), r(x) \in R[x]\)
(S2) \(p(x) + 0 = p(x) \quad \text{for all} \quad p(x) \in R[x]\)
(S3) For each \(p(x) \in R[x]\), there exists \(-p(x) \in R[x]\) such that \(p(x) + (-p(x)) = 0\)
(S4) \(p(x) + q(x) = q(x) + p(x) \quad \text{for all} \quad p(x), q(x) \in R[x]\)
(P1) \(\left[ p(x) q(x) \right] r(x) = p(x) \left[ q(x) r(x) \right] \quad \text{for all} \quad p(x), q(x), r(x) \in R[x]\)
(P2) \(p(x) \cdot 1 = p(x) \quad \text{for all} \quad p(x) \in R[x]\)
(P3) \(p(x)q(x) = q(x)p(x) \quad \text{for all} \quad p(x), q(x) \in R[x]\)
(D) \(p(x) \left[ q(x) + r(x) \right] = p(x)q(x) + p(x)r(x) \quad \text{for all} \quad p(x), q(x), r(x) \in R[x]\)
Corollary 1:
If \(p(x) \in \mathbb{C}[x]\), then \(p(x)\) is irreducible if and only if \(\deg(p(x)) = 1\).
Corollary 2:
Let \(p(x) \in \mathbb{C}[x]\) with \(\deg(p(x)) = n \geq 1\), then there exist \(C, C_1, \dots, C_n \in \mathbb{C}\) such that