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

$\forall f, g \in R \left[{X}\right]: \deg \left({f g}\right) \le \deg \left({f}\right) + \deg \left({g}\right)$


Degree of Product of Polynomials over Integral Domain not Less than Degree of Factors

$\forall f, g \in R \sqbrk X: \map \deg {f g} \ge \map \deg f$