Question

Am postat o poza cu un exercitiu cu polinoame pentru bac la mate . Va rog sa ma ajutati cu SUBPUNCTUL c) !

Answer 1

$f = X^4-3X^3+2X^2-X+1,\quad g = X^2-X+1 \\ \\ g(\alpha_1)+g(\alpha_2)+g(\alpha_3) +g(\alpha_4) = \\ \\ =\alpha_1^2-\alpha_1+1+\alpha_2^2-\alpha_2+1+\alpha_3^2-\alpha_3+1+\alpha_4^2-\alpha_4+1 = \\ \\ =\alpha_1^2+\alpha_2^2+\alpha_3^2+\alpha_4^2-(\alpha_1+\alpha_2+\alpha_3+\alpha_4)+4= \\ \\ =S_1^2 - 2S_2 - S_1+4 = \Big( -\dfrac{b}{a} \Big)^2-2\cdot \dfrac{c}{a} - \Big(- \dfrac{b}{a}\Big) +4 = \\ \\ = \Big(-\dfrac{-3}{1}\Big)^2-2\cdot \dfrac{2}{1}+ \dfrac{-3}{1} +4 = 9-4-3+4 = 12$

$\Rightarrow \boxed{g(\alpha_1)+g(\alpha_2)+g(\alpha_3) +g(\alpha_4) = 12}$