Properties of Degree
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Theorem
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$.
Let $R \sqbrk X$ denote the ring of polynomial forms over $R$ in the indeterminate $X$.
For $f \in R \sqbrk X$ let $\map \deg f$ denote the degree of $f$.
Then the following hold:
Degree of Sum of Polynomials
- $\forall f, g \in R \sqbrk X: \map \deg {f + g} \le \max \set {\map \deg f, \map \deg g}$
Degree of Product of Polynomials over Ring
- $\forall f, g \in R \sqbrk X: \map \deg {f g} \le \map \deg f + \map \deg g$
Degree of Product of Polynomials over Integral Domain not Less than Degree of Factors
- $\forall f, g \in R \sqbrk X: \map \deg {f g} \ge \map \deg f$