Question

Salut, imi poate explica cineva la 2 (a si c)? Va rog.

Answer 1

$f = (X^2-X+1)^{10}\\ \\ f = a_{20}X^{20}+a_{19}X^{19}+a_{18}X^{18}+...+a_{1}X+a_{0} \\ \\ a)~~f(1) = a_{20}+a_{19}+a_{18}+...+a_{0} \\ \Rightarrow (1-1+1)^{10} = a_{0}+a_{1}+a_{2}+...+a_{20}\\\\ \Rightarrow \boxed{a_{0}+a_{1}+a_{2}+...+a_{20} = 1} \\ \\ c)~~f = a_{0}+a_{1}X+a_{2}X^2+a_{3}X^3+a_{4}X^{4}+...+a_{20}X^{20} \\ \\ f(i) = a_0+a_1i-a_2-a_3i+a_4+...-a_{19}i+a_{20} \\ f(-i) = a_0-a_1i-a_2+a_3i+a_4+...+a_{19i}+a_{20} \\ \\ \text{Adunam:} \\ \\ f(i)+f(-i) = 2a_0-2a_2+2a_4+...+2a_{20}$

$-1+(-1) = 2(a_0-a_2+a_4-a_6+...+a_{20}) \\ 2(a_0-a_2+a_4-a_6+...+a_{20}) = -2\\ \\ \Rightarrow \boxed{a_0-a_2+a_4-a_6+...+a_{20} =-1}$