Polynomial's Common Multiple

Algebra Level 3

\[ \begin{eqnarray} f(x) &=& p(x)(x-1) +1 \\ f(x) &=& q(x)(x-2) +1 \\ f(x) &=& r(x)(x-3) +1 \end{eqnarray} \]

Let \(f(x)\) be a cubic monic polynomial, and \(p(x), q(x), r(x)\) be quadratic polynomials satisfying the system of equations above.