Solve xP(x-1)=(x-15)p(x)

$$x P ( x - 1 ) = ( x - 15 ) p ( x )$$

Solve for P (complex solution)

$$\left\{\begin{matrix}P=-\frac{p\left(15-x\right)}{x-1}\text{, }&x\neq 1\\P\in \mathrm{C}\text{, }&\left(p=0\text{ and }x=1\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Px^{2}-xP=\left(x-15\right)px$$

Use the distributive property to multiply x-15 by p.

$$Px^{2}-xP=\left(xp-15p\right)x$$

Use the distributive property to multiply xp-15p by x.

$$Px^{2}-xP=px^{2}-15px$$

Combine all terms containing P.

$$\left(x^{2}-x\right)P=px^{2}-15px$$

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

$$\frac{\left(x^{2}-x\right)P}{x^{2}-x}=\frac{px\left(x-15\right)}{x^{2}-x}$$

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

$$P=\frac{px\left(x-15\right)}{x^{2}-x}$$

Divide xp\left(-15+x\right) by x^{2}-x.

$$P=\frac{p\left(x-15\right)}{x-1}$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=\frac{P\left(x-1\right)}{x-15}\text{, }&x\neq 15\\p\in \mathrm{C}\text{, }&x=0\text{ or }\left(P=0\text{ and }x=15\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Px^{2}-xP=\left(x-15\right)px$$

Use the distributive property to multiply x-15 by p.

$$Px^{2}-xP=\left(xp-15p\right)x$$

Use the distributive property to multiply xp-15p by x.

$$Px^{2}-xP=px^{2}-15px$$

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

$$px^{2}-15px=Px^{2}-xP$$

Combine all terms containing p.

$$\left(x^{2}-15x\right)p=Px^{2}-xP$$

The equation is in standard form.

$$\left(x^{2}-15x\right)p=Px^{2}-Px$$

Divide both sides by x^{2}-15x.

$$\frac{\left(x^{2}-15x\right)p}{x^{2}-15x}=\frac{Px\left(x-1\right)}{x^{2}-15x}$$

Dividing by x^{2}-15x undoes the multiplication by x^{2}-15x.

$$p=\frac{Px\left(x-1\right)}{x^{2}-15x}$$

Divide xP\left(-1+x\right) by x^{2}-15x.

$$p=\frac{P\left(x-1\right)}{x-15}$$

Solve for P

$$\left\{\begin{matrix}P=-\frac{p\left(15-x\right)}{x-1}\text{, }&x\neq 1\\P\in \mathrm{R}\text{, }&\left(p=0\text{ and }x=1\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Px^{2}-xP=\left(x-15\right)px$$

Use the distributive property to multiply x-15 by p.

$$Px^{2}-xP=\left(xp-15p\right)x$$

Use the distributive property to multiply xp-15p by x.

$$Px^{2}-xP=px^{2}-15px$$

Combine all terms containing P.

$$\left(x^{2}-x\right)P=px^{2}-15px$$

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

$$\frac{\left(x^{2}-x\right)P}{x^{2}-x}=\frac{px\left(x-15\right)}{x^{2}-x}$$

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

$$P=\frac{px\left(x-15\right)}{x^{2}-x}$$

Divide xp\left(-15+x\right) by x^{2}-x.

$$P=\frac{p\left(x-15\right)}{x-1}$$

Solve for p

$$\left\{\begin{matrix}p=\frac{P\left(x-1\right)}{x-15}\text{, }&x\neq 15\\p\in \mathrm{R}\text{, }&x=0\text{ or }\left(P=0\text{ and }x=15\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Px^{2}-xP=\left(x-15\right)px$$

Use the distributive property to multiply x-15 by p.

$$Px^{2}-xP=\left(xp-15p\right)x$$

Use the distributive property to multiply xp-15p by x.

$$Px^{2}-xP=px^{2}-15px$$

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

$$px^{2}-15px=Px^{2}-xP$$

Combine all terms containing p.

$$\left(x^{2}-15x\right)p=Px^{2}-xP$$

The equation is in standard form.

$$\left(x^{2}-15x\right)p=Px^{2}-Px$$

Divide both sides by x^{2}-15x.

$$\frac{\left(x^{2}-15x\right)p}{x^{2}-15x}=\frac{Px\left(x-1\right)}{x^{2}-15x}$$

Dividing by x^{2}-15x undoes the multiplication by x^{2}-15x.

$$p=\frac{Px\left(x-1\right)}{x^{2}-15x}$$

Divide xP\left(-1+x\right) by x^{2}-15x.

$$p=\frac{P\left(x-1\right)}{x-15}$$