rút gọn biểu thức sau:x^30-2000x^29+2000x^28-2000x^27+...+2000x^2-2000x+2000 tại x=2006

$f(x)=x^{30}-2000x^{29}+2000x^{28}-2000x^{27}+...+2000x^2-2000x+2000$

$⇒f(2006)=2006^{30}-2000.2006^{29}+2000.2006^{28}-2000.2006^{27}+...+2000.2006^2-2000.2006+2000$

$⇒2006.f(2006)=2006^{31}-2000.2006^{30}+2000.2006^{29}-2000.2006^{28}+...+2000.2006^3-2000.2006^2+2000.2006$

$⇒2006.f(2006)+f(2006)=2006^{31}-2000.2006^{30}+2000.2006^{29}-2000.2006^{28}+...+2000.2006^3-2000.2006^2+2000.2006+2006^{30}-2000.2006^{29}+2000.2006^{28}-2000.2006^{27}+...+2000.2006^2-2000.2006+2000$

$⇒2007.f(2006)=2006^{31}-2000.2006^{30}+2006^{30}+2000.$

$⇒f(2006)=\frac{2006^{31}-2000.2006^{30}+2006^{30}+2000}{2007}$ 

$⇒f(2006)=\frac{2006^{30}(2006-2000+1)+2000}{2007}$ 

$⇒f(2006)=\frac{2006^{30}.7+2000}{2007}.$