Question

Exercitiile 31, 32, 33, 34, 35 va rog!

Answer 1

$P(X) = X^{10} \\ 32) ~~I(X) = X+1 \\ \\ P(X) = Q(X)\cdot I(X)+R \\ P(-1) = Q(-1)\cdot I(-1)+R \\ (-1)^{10} = 0+R \Rightarrow R = 1\\ \\ 33)~~I(X) = (X+1)^2 \\ \\ P(X) = Q(X)\cdot I(X)+aX+b \\ P(-1) = -a+b \Rightarrow b-a = 1 \\ \\ P'(X) = Q'(X)\cdot I(X)+Q(X)\cdot I'(X)+ a \\ P'(-1) = Q'(X)\cdot I(-1)+Q(X)\cdot I'(-1)+a \\ 10\cdot(-1)^9 = 0+0+a \Rightarrow a = -10 \Rightarrow b = -9 \Rightarrow D)~ \text{corect}$

$34)~~ I(X) = (X+1)^3 \\ \\ P(X) = Q(X)\cdot I(X)+ aX^2+bX+c \\ P(-1) = a-b+c \\ 1 = a-b+c\\ \\ P'(X) = Q'(X)\cdot I(X)+Q(X)\cdot I'(X)+2aX+b \\ P'(-1) = 0-2a+b \\ -10 = -2a+b \\ \\ P''(X) = Q''(X)\cdot I(X) +Q'(X)\cdot I'(X) + Q'(X)\cdot I''(X)+Q(X)\cdot I'(X)+\\+2a \\ \\ P''(-1) = 0+0+0+2a \Rightarrow 2a = 10\cdot 9\cdot (-1)^8 \Rightarrow 2a = 90 \Rightarrow a = 45 \\ -10=-90+b \Rightarrow b = 80 \Rightarrow c =1-45+80 = 36 \\ \\ \Rightarrow R(X) = 45X^2+80X+36 \Rightarrow B) \text{ corect}$