Ta có: \(x=2011\Rightarrow x+1=2012\)
Khi đó, ta có:
\(H\left(x\right)=x^4-\left(x+1\right).x^3+\left(x+1\right).x^2-\left(x+1\right).x+2012\)
\(=x^4-x^4-x^3+x^3+x^2-x^2-x+2012\)
\(\Rightarrow H\left(2011\right)=-2011+2012=1\).
Vậy \(H\left(2011\right)=1\)
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Cách 2:
\(H\left(x\right)=x^4-2012x^3+2012x^2-2012x+2012\)
\(=x^4-2011x^3-x^3+2011x^2+x^2-2011x-x+2011+1\)
\(=x^3\left(x-2011\right)-x^2\left(x-2011\right)+x\left(x-2011\right)-\left(x-2011\right)+1\)
\(=\left(x^3-x^2+x-1\right)\left(x-2011\right)+1\)
\(\Rightarrow H\left(2011\right)=1\)
Vậy...
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