\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(A, B, C \in \mathbb{K}[X]\) with \((A, B) \neq (0, 0)\). Consider the equation $$ A U + B V = C \qquad (U, V) \in \mathbb{K}[X]^2. \quad (\star) $$
  1. Show that \((\star)\) has a solution \(\iff (A \wedge B) \mid C\).
  2. Assume now that \(A, B \in \mathbb{K}[X]^*\) (both non-zero), that the solvability condition is satisfied, and write \(A = D A'\), \(B = D B'\) with \(D = A \wedge B\) and \(A' \wedge B' = 1\). Show that if \((U_0, V_0)\) is a particular solution, the set of all solutions is $$ \{ (U_0 - B' T, \; V_0 + A' T) : T \in \mathbb{K}[X] \}. $$
  3. Apply this to find all \((U, V) \in \mathbb{R}[X]^2\) satisfying \((X^2 - 1) U + (X^2 + X) V = X + 1\).

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