Find all non-zero polynomials satisfying the equation: x*W(x+1) = (x+2)*W(x) x ∊ R.

Question

W(x)/W(x+1)=x/(x+2)

Answer 1

One possible solution is the polynomial \(W(x) = kx\) where \(k \neq 0\) is a constant.

To show that this is a solution, we can plug this into the equation:

\[ \frac{kx}{k(x+1)} = \frac{x}{x+2} \]

\[ \frac{x}{x+1} = \frac{x}{x+2} \]

\[ x(x+2) = x(x+1) \]

which is true for all real numbers \(x\).

Therefore, the polynomial \(W(x) = kx\) where \(k \neq 0\) is a solution to the equation \(xW(x+1) = (x+2)W(x)\).