Question

sa se rezolve exercitiul:

Answer 1

$P(3) = 2\\ P(-2) = 3\\ \\ R(X) = aX+b\\ \\P(3X+2) = Q(X)\cdot (9X^2+9X-4)+R(X)\\ \\ 3X+2 = 3 \Rightarrow X = \dfrac{1}{3} \\ \\ P(3) = Q\Big(\dfrac{1}{3}\Big)\cdot \Big(9\cdot \dfrac{1}{9}+9\cdot \dfrac{1}{3}}-4\Big)+\dfrac{a}{3}+b \\ 2 = Q\Big(\dfrac{1}{3}\Big)\cdot 0+\dfrac{a}{3}+b \\ 6 = a+3b \\ \\ 3X+2 = -2 \Rightarrow X=-\dfrac{4}{3} \\ \\ P(-2)=Q\Big(-\dfrac{4}{3}\Big)\cdot \Big(9\cdot \dfrac{16}{9}-9\cdot \dfrac{4}{3}-4\Big)-\dfrac{4a}{3}+b$

$3 = Q\Big(-\dfrac{4}{3}\Big)\cdot 0-\dfrac{4a}{3}+b \\\\ 9 = -4a+3b \\ \\ 6 = a+3b\\ \\ \Rightarrow 3 = -5a \Rightarrow a = -\dfrac{3}{5} \\ \\ 3b = 6+\dfrac{3}{5} = \dfrac{33}{5} \Rightarrow b = \dfrac{33}{15}\\ \\ \Rightarrow \boxed{R(X) = -\dfrac{3}{5}X+\dfrac{33}{15}}$