Question

4. Fie P(x) un polinom de gradul n .... continuarea în imagine.


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Answer 1

$P(x) = \dfrac{a}{n!}(x-c)^n \\ \\\text{Logaritmam:} \\ \\ \ln\Big(P(x)\Big) = \ln\Big(\dfrac{a}{n!}\cdot (x-c)^n\Big) \\ \\ \ln\Big(P(x)\Big) = \ln \Big(\dfrac{a}{n!}\Big)+\ln (x-c)^n \\ \\ \ln\Big(P(x)\Big) = \ln \Big(\dfrac{a}{n!}\Big)+n\cdot \ln(x-c)\\ \\ \text{Derivam: }\\ \\ \ln\Big(P(x)\Big)' = 0+n\cdot \dfrac{(x-c)'}{x-c} \\ \\ \dfrac{P'(x)}{P(x)} = \dfrac{n}{x-c} \\ \\ \dfrac{P(x)}{P'(x)} = \dfrac{x-c}{n}\Rightarrow P(x) = \Big(\dfrac{x-c}{n}\Big)\cdot P'(x)+0\\ \\ \Rightarrow P(x) \text{ se divide prin }P'(x)$