Question

Chứng minh rằng đa thức P(x)= x^2017+x^2+1 chia hết cho đa thức Q(x)= x^2+x+1

Answer 1

\(P\left(x\right)=x^{2017}+x^2+1\)

\(=\left(x^{2017}-x\right)+\left(x^2+x+1\right)\)

\(=x\left(x^{2016}-1\right)+\left(x^2+x+1\right)\)

\(=x\left[\left(x^3\right)^{2016}-1\right]+\left(x^2+x+1\right)\)

\(=x\left(x^3-1\right)A+\left(x^2+x+1\right)\)

\(=x\left(x-1\right)\left(x^2+x+1\right)A+\left(x^2+x+1\right)\)

\(A=\left(x^2+x+1\right)\left[x\left(x-1\right)A+1\right]⋮x^2+x+1\) (đpcm)