Smallest polynomial \(f\) s.t. \(72 | f(x)\) if \(x\) is an integer

Find the polynomial \(f\) with the following properties:

• its leading coefficient is 1.

• its coefficents are nonnegative integers.

• \(72 | f(x)\) if \(x\) is an integer.

• if \(g\) is another polynomial with the same properties, then \(g-f\) has a nonnegative leading coefficient.

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Partial Idea

We start with \(x^6 + x^4 -2x^2 = x^2(x^2+2)(x^2-1)\) which is always divisible by \(72\) when \(x\) is an integer greater than 1.

Adding \(36x^2 + 36x = 36x(x+1)\), we get \(x^6 + x^4 + 34x^2 + 36x\) which is also divisible by \(72\) when \(x\) is an integer.

Note that both the even part and odd part are divisbile by \(36\).

Official Solution (Shared by 申强老师）