-2 to pierwiastek, więc W(-2)=0
[tex]W(-2) = - 8 + 4m - 2n + 2p - 8 \\ \\ - 16 + 4m - 2n + 2p = 0 \\ \\ - 8 + 2m - n + p = 0 \\ [/tex]
Liczby 1, m,n,p tworzą ciąg geometryczny, więc :
[tex] \frac{m}{1} = \frac{n}{m} = \frac{p}{n} \\ \\ \frac{m}{1} = \frac{n}{m}\rightarrow n = {m}^{2} \\ \\ \frac{n}{m} = \frac{p}{n}\rightarrow {n}^{2} = pm \\[/tex]
[tex]- 8 + 2m - n + p = 0 \\ \\ p = 8 - 2m + n \\ [/tex]
[tex] {n}^{2} = pm \\ \\ { {m}^{2} }^{2} = (8 - 2m + n)m \\ \\ {m}^{4} = 8m - 2 {m}^{2} + {m}^{3} \\ \\ {m}^{4} - {m}^{3} + 2 {m}^{2} - 8m = 0 \\ \\ m( {m}^{2} + m + 4)(m - 2) = 0 \\ \\ m = 0 \ \vee {m}^{2} + m + 4 = 0 \ \vee m - 2 = 0 \\ \\ m = 0 \ \vee m = 2 \\ \\ \frac{m}{1} = \frac{n}{m} = \frac{p}{n} \rightarrow D: m \wedge n \neq0 \\ \\ \boxed{m=2} \\ [/tex]
[tex] {m}^{2} = n \\ n = 4 \\ [/tex]
[tex]p = 8 - 2 \times 2 + 4 \\ p = 8 \\ [/tex]
[tex]W(x) = {x}^{3} + 2 {x}^{2} + 4x + 16 - 8 \\ W(x) = {x}^{3} + 2 {x}^{2} + 4x + 8 \\ [/tex]
