Question

Se considera polinomul f=(x-1)^{2017} +(x-2)^{2007}

b) Determinați restul împărţirii polinomului f la polinomul g=x^2-3x+2.
c) Calculați suma rădăcinilor polinomului f .

Va rog daca pute-ti sa ma ajuta-ti

Answer 1

$f = (x-1)^{2017}+(x-2)^{2017}\\ \\ g = x^2-3x+2 = (x-1)(x-2)$

$\\\\b)\,\,\, \text{grad }r < \text{grad }g \Rightarrow \text{grad }r < 2 \Rightarrow R(X) = aX+b\\ \\f(X) = C(X)\cdot g(X)+R(X)\\\\ f(1)=C(1)\cdot g(1)+R(1)\\ \\ \Rightarrow (1-1)^{2017}+(1-2)^{2017} = C(1)\cdot 0+a+b\\ \\ \Rightarrow 0-1 = 0+a+b \Rightarrow a+b=-1\\ \\ f(2) = C(2)\cdot g(2)+R(2)\\ \\ \Rightarrow (2-1)^{2017}+(2-2)^{2017} = C(2)\cdot 0+2a+b\\ \\ \Rightarrow 1+0 = 0+2a+b\Rightarrow 2a+b = 1\\ \\ \Rightarrow a = 2\Rightarrow b = -3\Rightarrow \boxed{r = 2x-3}$

$\\\\c)\,\,\,T_{k+1} = C_{n}^k\cdot a^{n-k}\cdot b^k\\ \\ (x-1)^{2017}+(x-2)^{2017} = \\ \\ =x^{2017}+C_{2017}^{1}\cdot x^{2017-1}\cdot (-1)^{1}+...+x^{2017}+C_{2017}^{1}\cdot x^{2017-1}\cdot (-2)^1+...\\ \\ = 2x^{2017}-(C_{2017}^1+2C_{2017}^1)x^{2016}+...\\ \\ = 2x^{2017}-(2017+2\cdot 2017)x^{2016}+...\\ \\ =2x^{2017}-6051x^{2016}+...\\ \\ \Rightarrow S =-\dfrac{-6051}{2} \Rightarrow \boxed{S = \dfrac{6051}{2}}$