Solve (x)=q(x)*(x^2+1)+r(x)

$$( x ) = q ( x ) \cdot ( x ^ { 2 } + 1 ) + r ( x )$$

Solve for q

$$\left\{\begin{matrix}\\q=-\frac{r-1}{x^{2}+1}\text{, }&\text{unconditionally}\\q\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply qx by x^{2}+1.

$$x=qx^{3}+qx+rx$$

Swap sides so that all variable terms are on the left hand side.

$$qx^{3}+qx+rx=x$$

Subtract rx from both sides.

$$qx^{3}+qx=x-rx$$

Combine all terms containing q.

$$\left(x^{3}+x\right)q=x-rx$$

Divide both sides by x^{3}+x.

$$\frac{\left(x^{3}+x\right)q}{x^{3}+x}=\frac{x-rx}{x^{3}+x}$$

Dividing by x^{3}+x undoes the multiplication by x^{3}+x.

$$q=\frac{x-rx}{x^{3}+x}$$

Divide x-xr by x^{3}+x.

$$q=\frac{1-r}{x^{2}+1}$$

Solve for r

$$\left\{\begin{matrix}\\r=1-q-qx^{2}\text{, }&\text{unconditionally}\\r\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply qx by x^{2}+1.

$$x=qx^{3}+qx+rx$$

Swap sides so that all variable terms are on the left hand side.

$$qx^{3}+qx+rx=x$$

Subtract qx^{3} from both sides.

$$qx+rx=x-qx^{3}$$

Subtract qx from both sides.

$$rx=x-qx^{3}-qx$$

Reorder the terms.

$$rx=-qx^{3}-qx+x$$

The equation is in standard form.

$$xr=x-qx-qx^{3}$$

Divide both sides by x.

$$\frac{xr}{x}=\frac{x-qx-qx^{3}}{x}$$

Dividing by x undoes the multiplication by x.

$$r=\frac{x-qx-qx^{3}}{x}$$

Divide -qx^{3}-xq+x by x.

$$r=1-q-qx^{2}$$