Descarte’s rule of signs

For a polynomial $𝑓 (𝑥) \in ℝ [𝑥]$ , let $c (𝑓 (𝑥))$ be the number of sign changes in the sequence of coëfficients, and let $r (𝑓 (𝑥))$ be the number of real roots. anal Then

r (𝑓 (𝑥)) = c (𝑓 (𝑥)) - 2 𝑘

for some $𝑘 \in ℕ 0$ .

Proof

proof

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Descarte’s rule of signs

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