From USSR with love for algebra !

Problem:

Find non-zero distinct integers a, b and c such that the following fourth-degree polynomial with integral coefficients, can be written as the product of two other polynomials with integral coefficients:

x(x-a)(x-b)(x-c)+1.

Solution: TBD.

Hint: Of course, the natural thought that comes to mind is the method of undetermined coefficients, and perhaps, some more trial and error to obtain some coefficients, and if we find these, we can find the remaining through their relationship with other coefficients. That brings to mind, Viete’s relations also !! But, there will be two cases here: in one case, the polynomial is a product of two quadratic polynomials, and in the other case, the given fourth degree polynomial is a product of a first degree polynomial and a third degree polynomial.

Problem:

For which integers a_{1}, a_{2}, \ldots, a_{n}, where these are all distinct, are the following polynomials with integral coefficients expressible as the product of two polynomials with integral coefficients?

(a) (x-a_{1})(x-a_{2})(x-a_{3}) \ldots (x-a_{n}) -1

(b) (x-a_{1})(x-a_{2})(x-a_{3}) \ldots (x-a_{n}) + 1

Solution: TBD.

Hint: This is seems to be related to the previous question of fourth degree polynomial ! But, I am not sure if my hint will work ! No spoon-feeding please ! Well, that said, I will think of a clever hint in a couple of days.

Problem:

Prove that if the integers a_{1}, a_{2}, \ldots, a_{n} are all distinct, then the polynomial

(x-a_{1})^{2}(x-a_{2})^{2}\ldots (x-a_{n})^{2} + 1

cannot be expressed as a product of two other polynomials with integral coefficients.

Solution: TBD.

Remark: What can be said if the polynomial were

(x-a_{1})^{2}(x-a_{2})^{2}\ldots (x-a_{n})^{2} -1

Remark: It is too premature right now for math olympiad students, but polynomial factorization/decomposition has applications in a practical field called control systems.

More later,

Nalin Pithwa

 

 

 

 

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