I am trying to prove the following problem:
Prove that there's no such polynomial $p(x)$ with integer coefficients, such that $p(a) = p(b) = $ $p(c) = p(d) = 4$ and $p(e) = 10$, where $a, b, c, d, e$ are integers themselves and are distinct.
If $p(x)$ is the polynomial and $p(a) = p(b) = p(c) = p(d) = 4$ then it has the form $p(x) = (x-a)(x-b)(x-c)(x-d)+4.$
Now I can't figure out how I prove that there is no such integer $e$ that $p(e) = 10$.
I tried constructing various polynomials in Mathematica (as Lagrange Interpolating polynomial) and I always ended up having something like $p(x) = \frac{(x-a)(x-b)(x-c)(x-d)}{\text{const}}+4$ for a polynomial that interpolates points $(a,4), (b,4), (c,4), (d,4), (e,10)$.
I can't find a good argument that the polynomial $(x-a)(x-b)(x-c)(x-d)+4$ will always be divided by some $\text{const}$ to meet the $p(e)=10$ requirement, therefore there is no such polynomial with integer coefficients that.
Can anyone help?