Solve x^3/(x-7)(x+1)=f(x)

$$\frac { x ^ { 3 } } { ( x - 7 ) ( x + 1 ) } = f ( x )$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{x^{2}}{\left(x-7\right)\left(x+1\right)}\text{, }&x\neq 7\text{ and }x\neq -1\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-7\right)\left(x+1\right).

$$x^{3}=fx\left(x-7\right)\left(x+1\right)$$

Use the distributive property to multiply fx by x-7.

$$x^{3}=\left(fx^{2}-7fx\right)\left(x+1\right)$$

Use the distributive property to multiply fx^{2}-7fx by x+1 and combine like terms.

$$x^{3}=fx^{3}-6fx^{2}-7fx$$

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

$$fx^{3}-6fx^{2}-7fx=x^{3}$$

Combine all terms containing f.

$$\left(x^{3}-6x^{2}-7x\right)f=x^{3}$$

Divide both sides by x^{3}-6x^{2}-7x.

$$\frac{\left(x^{3}-6x^{2}-7x\right)f}{x^{3}-6x^{2}-7x}=\frac{x^{3}}{x^{3}-6x^{2}-7x}$$

Dividing by x^{3}-6x^{2}-7x undoes the multiplication by x^{3}-6x^{2}-7x.

$$f=\frac{x^{3}}{x^{3}-6x^{2}-7x}$$

Divide x^{3} by x^{3}-6x^{2}-7x.

$$f=\frac{x^{2}}{\left(x-7\right)\left(x+1\right)}$$

Solve for f

$$\left\{\begin{matrix}f=\frac{x^{2}}{\left(x-7\right)\left(x+1\right)}\text{, }&x\neq 7\text{ and }x\neq -1\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-7\right)\left(x+1\right).

$$x^{3}=fx\left(x-7\right)\left(x+1\right)$$

Use the distributive property to multiply fx by x-7.

$$x^{3}=\left(fx^{2}-7fx\right)\left(x+1\right)$$

Use the distributive property to multiply fx^{2}-7fx by x+1 and combine like terms.

$$x^{3}=fx^{3}-6fx^{2}-7fx$$

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

$$fx^{3}-6fx^{2}-7fx=x^{3}$$

Combine all terms containing f.

$$\left(x^{3}-6x^{2}-7x\right)f=x^{3}$$

Divide both sides by x^{3}-6x^{2}-7x.

$$\frac{\left(x^{3}-6x^{2}-7x\right)f}{x^{3}-6x^{2}-7x}=\frac{x^{3}}{x^{3}-6x^{2}-7x}$$

Dividing by x^{3}-6x^{2}-7x undoes the multiplication by x^{3}-6x^{2}-7x.

$$f=\frac{x^{3}}{x^{3}-6x^{2}-7x}$$

Divide x^{3} by x^{3}-6x^{2}-7x.

$$f=\frac{x^{2}}{\left(x-7\right)\left(x+1\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{f\left(16f-7\right)}-3f}{1-f}\text{; }x=-\frac{\sqrt{f\left(16f-7\right)}+3f}{1-f}\text{, }&f\neq 1\\x=-\frac{7}{6}\text{, }&f=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{f\left(16f-7\right)}-3f}{1-f}\text{; }x=-\frac{\sqrt{f\left(16f-7\right)}+3f}{1-f}\text{, }&f\leq 0\text{ or }\left(f\neq 1\text{ and }f\geq \frac{7}{16}\right)\\x=-\frac{7}{6}\text{, }&f=1\end{matrix}\right.$$