Solve (x-1)*P(x)=x^7+x^5+kquadP(1)=?

$$( x - 1 ) \cdot P ( x ) = x ^ { 7 } + x ^ { 5 } + k \quad P ( 1 ) = ?$$

Solve for P

$$\left\{\begin{matrix}P=\frac{\left(x^{2}+1\right)x^{5}}{x^{2}-x-k}\text{, }&k\neq x\left(x-1\right)\\P\in \mathrm{R}\text{, }&x=0\text{ and }k=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by P.

$$\left(xP-P\right)x=x^{7}+x^{5}+kP\times 1$$

Use the distributive property to multiply xP-P by x.

$$Px^{2}-Px=x^{7}+x^{5}+kP\times 1$$

Subtract kP\times 1 from both sides.

$$Px^{2}-Px-kP\times 1=x^{7}+x^{5}$$

Reorder the terms.

$$Px^{2}-Px-Pk=x^{7}+x^{5}$$

Combine all terms containing P.

$$\left(x^{2}-x-k\right)P=x^{7}+x^{5}$$

Divide both sides by x^{2}-x-k.

$$\frac{\left(x^{2}-x-k\right)P}{x^{2}-x-k}=\frac{x^{7}+x^{5}}{x^{2}-x-k}$$

Dividing by x^{2}-x-k undoes the multiplication by x^{2}-x-k.

$$P=\frac{x^{7}+x^{5}}{x^{2}-x-k}$$

Divide x^{7}+x^{5} by x^{2}-x-k.

$$P=\frac{\left(x^{2}+1\right)x^{5}}{x^{2}-x-k}$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{x\left(x^{6}+x^{4}-Px+P\right)}{P}\text{, }&P\neq 0\\k\in \mathrm{R}\text{, }&x=0\text{ and }P=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by P.

$$\left(xP-P\right)x=x^{7}+x^{5}+kP\times 1$$

Use the distributive property to multiply xP-P by x.

$$Px^{2}-Px=x^{7}+x^{5}+kP\times 1$$

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

$$x^{7}+x^{5}+kP\times 1=Px^{2}-Px$$

Subtract x^{7} from both sides.

$$x^{5}+kP\times 1=Px^{2}-Px-x^{7}$$

Subtract x^{5} from both sides.

$$kP\times 1=Px^{2}-Px-x^{7}-x^{5}$$

Reorder the terms.

$$Pk=-x^{7}-x^{5}+Px^{2}-Px$$

Divide both sides by P.

$$\frac{Pk}{P}=\frac{x\left(-x^{6}-x^{4}+Px-P\right)}{P}$$

Dividing by P undoes the multiplication by P.

$$k=\frac{x\left(-x^{6}-x^{4}+Px-P\right)}{P}$$