find a non-constant polynomial function $p(x)$ such that $p(1)=p(2)=p(3)=4$. I try to solve it but do not know where to start (every time I substitute a number into the equation, I get three more unknown numbers). Would anyone give me any clue? Is there a general method to deal with this kind of problems?
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Hint: Any polynomial of shape $(x-1)(x-2)(x-3)q(x)$ will be $0$ at $x=1,2,3$.
Remark: Alternately, we can start as you did, and try to find a cubic $ax^3+bx^2+cx+d$ that will work. We get $3$ linear equations in $4$ unknowns, to which we can find a solution. Straightforward, but more calculation.
You asked for a general method. The idea of the hint above generalizes to Lagrange Interpolation. If you want to use the machinery of Lagrange Interpolation, to make sure it does not give you the trivial answer $p(x)=4$, you can specify, for example, that $p(0)=0$.