Question

Exercitiul 5, cu EXPLICATII, va rog!

Answer 1

$Q(X) = (X^2+3X+2)^{2016}+mX^3+2nX+4\\ \\ Q(X) = C(X)\cdot (X+1)^2\\ \\ \boxed{1}\,\,\,\, Q(-1) =C(-1)\cdot (-1+1)^2 \\ \\ \Rightarrow Q(-1) = 0\\ \\\Rightarrow \left[1+3\cdot (-1)+2\right]^{2016} +m\cdot(-1)^3+2n\cdot (-1)+4 = 0\\ \\ \Rightarrow 0-m-2n = -4\Rightarrow m+2n = 4\\ \\ \boxed{2}\,\,\, Q'(X)= C'(X)\cdot (X+1)^2+C(X)\cdot \left[(X+1)^2\right]'\\ \\ Q'(X) = C'(X)\cdot (X+1)^2+2C(X)\cdot (X+1)\\ \\\text{Fac }x = -1:\\Q'(-1) = C'(-1)\cdot 0+2C(-1)\cdot 0 = 0$

$\text{Dar }Q'(X) = 2016(X^2+3X+2)^{2015}\cdot(2X+3)+3mX^2+2n\\ \\ \Rightarrow 2016\cdot 0\cdot 1+3m\cdot (-1)^2+2n = 0 \\ \\ \Rightarrow 3m+2n = 0\Rightarrow 2n = -3m\\ \\ \text{Din (1) si (2):} \\m-3m = 4 \Rightarrow -2m = 4 \Rightarrow \boxed{m = -2}\\ \Rightarrow 2n = 6 \Rightarrow \boxed{n = 3}\\ \\ \textbf{Raspuns:}\\ a)\,\,\,p = -6$