Degree of a polynomial

Let $𝑓 (𝑥) \in 𝑅 [𝑥]$ be a polynomial. The degree $d e g 𝑓 (𝑥)$ is the largest exponent of $𝑥$ with nonzero coëfficient in $𝑓 (𝑥)$ . This makes $𝑅 [𝑥]$ a $ℕ - \infty$ -graded ring, since

d e g (𝑓 (𝑥) \cdot 𝑔 (𝑥)) = (d e g 𝑓 (𝑥)) (d e g 𝑔 (𝑥)) .

Properties

If $𝑅$ is an integral domain and $𝑓 (𝑥)$ has $𝑛$ distinct solutions, then $d e g 𝑓 (𝑥) \geq 𝑛$ .

Proof of 1

Let $𝑓 (𝑥) \in 𝑅 [𝑥]$ and suppose $𝑎 1, \dots, 𝑎 𝑛$ are distinct solutions of $𝑓 (𝑥)$ . It follows from the polynomial factor theorem that

(𝑥 - 𝑎 1) \dots (𝑥 - 𝑎 𝑛) ∣ 𝑓 (𝑥)

whence $𝑓 (𝑥)$ has degree at least $𝑛$ , proving ^P1,

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