This question is related to this:https://math.stackexchange.com/questions/1412564/interpolating-polynomial-its-root
We have $P_3=P_2\cdot P_1$,for three non-zero polynomials. The degree of each polynomial is at least 1.
Question: Does $P_1$ overlap $P_3$ over more than one point?
By overlap I do not mean intersect. Instead, I mean a part of the polynomials can lay on each other over more than one point. edit: $P_1\neq P_2$
If two polynomials agree on any open interval (which seems to be what you mean by "lay on each other over more than one point"), then they are identical polynomials.
So the answer to your question is "no", because $deg(P_3) \ge 1 + deg(P_2)$, hence $P_3$ cannot be a constant multiple of $P_1$.
Why must the two polynomials be equal?
Consider points $x = x_0, x_1, \ldots, x_n$, where $n$ is the higher of the degrees of the two polynomials, and the $x_i$ are distinct and lie in the shared interval.
If you plug these into both polynomials, you get $$ p_1(x_0) = p_3(x_0)\\ p_1(x_1) = p_3(x_1) \\ \ldots \\ p_1(x_n) = p_3(x_n) $$ which gives you $n$ linear equations in the numbers $c_i = a_i - b_i$, where the $a_i$ and $b_i$ are the coefficients of the two polynomials. These can be written in matrix form as $$ V \begin{bmatrix} c_1 \\ c_2 \\ \ldots \\ c_n \end{bmatrix} = 0, $$ where $V$ is a Vandermonde matrix, whose determinant is known to be nonzero, so we can conclude that all $c_i$ are zero, so that the coefficients of the two polynomials are identical.