Solve f(x)+g(x)=x(f+g)

$$f ( x ) + g ( x ) = x ( f + g )$$

Solve for f (complex solution)

$$f\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply x by f+g.

$$fx+gx=xf+xg$$

Subtract xf from both sides.

$$fx+gx-xf=xg$$

Combine fx and -xf to get 0.

$$gx=xg$$

Reorder the terms.

$$\text{true}$$

This is true for any f.

$$f\in \mathrm{C}$$

Solve for g (complex solution)

$$g\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply x by f+g.

$$fx+gx=xf+xg$$

Subtract xg from both sides.

$$fx+gx-xg=xf$$

Combine gx and -xg to get 0.

$$fx=xf$$

Reorder the terms.

$$\text{true}$$

This is true for any g.

$$g\in \mathrm{C}$$

Solve for f

$$f\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply x by f+g.

$$fx+gx=xf+xg$$

Subtract xf from both sides.

$$fx+gx-xf=xg$$

Combine fx and -xf to get 0.

$$gx=xg$$

Reorder the terms.

$$\text{true}$$

This is true for any f.

$$f\in \mathrm{R}$$

Solve for g

$$g\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply x by f+g.

$$fx+gx=xf+xg$$

Subtract xg from both sides.

$$fx+gx-xg=xf$$

Combine gx and -xg to get 0.

$$fx=xf$$

Reorder the terms.

$$\text{true}$$

This is true for any g.

$$g\in \mathrm{R}$$