AL 278*
[tex]P(X) = Q(X) + Q(1-X) \\ \\ P(0) = Q(0) + Q(1) = 0 \\ \\ P(1) = Q(1)+Q(0) = 0 \\ \\ P(X) = a_{n}X^n + a_{n-1}X^{n-1}+...+a_1X+a_0 \\ \\ P(1) = 0 \Rightarrow a_{n}+a_{n-1}+...+a_{0} = 0 \\ \\ \text{Dar } a_n, a_{n-1},...,a_0 \text{ sunt naturali, }\geq 0\Rightarrow \\ a_{n},a_{n-1},...,a_{0} = 0 \Rightarrow P(X) = 0 \\ \\ \Rightarrow P(P(3)) = 0 \Rightarrow a) \text{ corect}[/tex]
AL 279*
[tex]P(X) = -P(-X) \\ \text{Restul impartirii lui P la X-7 este 3}\Rightarrow \\ \\ \Rightarrow P(7) = 3 \\ \\ P(0) = -P(0)\Rightarrow 2P(0) = 0 \Rightarrow P(0) = 0\\\\P(X) = Q(X)\cdot C(X) + R(X) \\ P(X) = (X^2-7X)\cdot C(X) + R(X) \\ P(X) = X(X-7)\cdot C(X)+R(X) \\ R(X) \text{ are cel mult gradul 1} \Rightarrow R(X) = aX+b \\ \\ P(7) = 0+R(7) = 7a+b = 3\\ P(0) = 0+R(0) = b = 0[/tex]
[tex]\Rightarrow 7a = 3 \Rightarrow a = \dfrac{3}{7} \Rightarrow \\ \Rightarrow R(X) = \dfrac{3}{7}\cdot X [/tex]
