\(P\left(x\right)=x^5-2013x^4+2013x^3-2013x^2+2013x-2014\)
\(=x^5-2012x^4-x^4+2012x^3+x^3-2012x^2-x^2+2012x+x-2014\)
\(=\left(x^5-x^4\right)+\left(-2012x^4+2012x^3\right)+\left(x^3-x^2\right)+\left(-2012x^2+2012x\right)+x-2014\)
\(=x^4\left(x-1\right)-2012x^3\left(x-1\right)+x^2\left(x-1\right)-2012x\left(x-1\right)+\left(x-1\right)-2013\)
\(=\left(x-1\right)\left(x^4-2012x^3+x^2-2012x+1\right)-2013\)
\(=\left(x-1\right)\left(x^3\left(x-2012\right)+x\left(x-2012\right)+1\right)-2013\)
Thay x=2012 ta có :
\(P\left(x\right)=\left(2012-1\right)\left(2012^3\left(20112-2012\right)+2012\left(2012-2012\right)+1\right)-2013\)
\(=2011\left(2012^3\cdot0+2012\cdot0+1\right)-2013\)
\(=2011\cdot\left(1\right)-2013\\ =-2\)
\(P\left(x\right)=x^5-\left(2012+1\right)x^4+\left(2012+1\right)x^3-\left(2012+1\right)x^2+\left(2012+1\right)x-\left(2012+2\right)\)
\(=x^5-\left(x+1\right)x^4+\left(x+1\right)x^3-\left(x+1\right)x^2+\left(x+1\right)x-\left(x+2\right)\)
\(=x^5-x^5-x^4+x^4+x^3-x^3-x^2+x^2+x-x-2\)
\(\Rightarrow P\left(x\right)=-2\)
\(P\left(x\right)=x^5-\left(2012+1\right)x^4+\left(2012+1\right)x^3-\left(2012+1\right)x^2+\left(2012+1\right)x-\left(2012+2\right)\)
\(P\left(x\right)=x^5-\left(x+1\right)x^4+\left(x+1\right)x^3-\left(x+1\right)x^2+\left(x+1\right)x-\left(x+2\right)\)
\(P\left(x\right)=x^5-x^5-x^4+x^4+x^3-x^3-x^2+x^2+x-x-2\)
\(\Rightarrow P\left(x\right)=2\)
