Question

Tìm dư khi chia \(x+x^3+x^9+x^{27}\) cho \(x^2-1\)

Answer 1

\(P\left(x\right)=x^{27}+x^9+x^3+x\)

\(Q\left(x\right)=x^2-1\)

Do Q(x) bậc 2 nên số dư cao nhất là bậc, 1 giả sử \(P\left(x\right)=Q\left(x\right).R\left(x\right)+ax+b\)

\(\Leftrightarrow x^{27}+x^9+x^3+x=\left(x^2-1\right)R\left(x\right)+ax+b\)

Thay \(x=1\Rightarrow4=a+b\)

Thay \(x=-1\Rightarrow-4=-a+b\)

\(\Rightarrow\left\{{}\begin{matrix}a+b=4\\-a+b=-4\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=4\\b=0\end{matrix}\right.\) \(\Rightarrow\) P(x) chia Q(x) dư \(4x\)