a)

Wykonamy dzielenie  $W\left(x\right):P\left(x\right)=\left(x^3-10x^2+2x+7\right):\left(x-1\right)$ sposobem pisemnym.

$\begin{array}{l}{x}^2-9x-7\\ \overline{\left(x^3-10x^2+2x+7\right)}:\left(x-1\right)\\ \underline{-x^3+x^2}\\ -9x^2+2x+7\\ \underline{-9x^2-9x}\\ -7x+7\\ \underline{7x-7}\\ 0\end{array}$

Zatem  $\left(x^3-10x^2+2x+7\right):\left(x-1\right)=x^2-9x-7.$

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b)

Wykonamy dzielenie $W\left(x\right):P\left(x\right)=\left(2x^3+x^2-x+10\right):\left(x+2\right)$ sposobem pisemnym.

$\begin{array}{l}2x^2-3x+5\\ \overline{\left(2x^3+x^2-x+10\right)}:\left(x+2\right)\\ \underline{-2x^3-4x^2}\\ -3x^2-x+10\\ \underline{-3x^2+6x}\\ 5x+10\\ \underline{-5x-10}\\ 0\end{array}$

Zatem $\left(2x^3+x^2-x+10\right):\left(x+2\right)=2x^2-3x+5.$

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c)

Wykonajmy dzielenie $W\left(x\right):P\left(x\right)=\left(x^4+3x^3-4x^2\right):\left(x+1\right)$ sposobem pisemnym.

$\begin{array}{l}{x}^3+2x^2-6x+1\\ \overline{\left(x^4+3x^3-4x^2-5x+1\right)}:\left(x+1\right)\\ \underline{-x^4-x^3}\\ 2x^3-4x^2-5x+1\\ \underline{-2x^3-2x^2}\\ -6x^2-5x+1\\ \underline{6x^2+6x}\\ x+1\\ \underline{-x-1}\\ 0\end{array}$

Zatem $\left(x^4+3x^3-4x^2-5x+1\right):\left(x+1\right)=x^3+2x^2-6x+1.$

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d)

Zauważmy, że $x^3-6x+4=x^3+0-6x+4.$

Wykonajmy dzielenie $W\left(x\right):P\left(x\right)=\left(x^3+0-6x+4\right):\left(x-2\right)$ sposobem pisemnym.

 $\begin{array}{l}{x}^2-9x-7\\ \overline{\left(x^3+0-6x+4\right)}:\left(x-2\right)\\ \underline{-x^3+2x^2}\\ 2x^2-6x+4\\ \underline{-2x^2+4x}\\ -2x+4\\ \underline{2x-4}\\ 0\end{array}$

Zatem $\left(x^3-6x+4\right):\left(x-2\right)=x^2+2x-2.$