Question

Answer 1

\(A=2x^{43}-3x^{35}+\left(x-1\right)^2\)

\(=2x^{43}-3x^{35}+x^2-2x+1\)

\(=2x\left(x^{42}-1\right)-x^2\left(x^{33}-1\right)-2x^{35}+1\)

\(=2x\left(x^{42}-1\right)-x^2\left(x^{33}-1\right)-2x^{32}\left(x^3-1\right)-2x^{29}\left(x^3-1\right)-...-2x^2\left(x^3-1\right)-2\left(x^2+x+1\right)+2x+3\)

Ta có tính chất: \(\left(x^{3m}-1\right)⋮\left(x^3-1\right)\left(m\in N\cdot\right)\) (áp dụng hệ quả nhị thức Newton) và \(\left(x^3-1\right)⋮\left(x^2+x+1\right)\).

\(\Rightarrow A\) chia \(x^2+x+1\) dư \(2x+3\)