XLV OM - II - Task 1

Determine all polynomials $ P(x) $ of degree at most fifth with real coefficients, having the property that the polynomial $ P(x) + 1 $ is divisible by $ (x - 1)^3 $ and the polynomial $ P(x) - 1 $ is divisible by $ (x + 1)^3 $.
Suppose that the polynomial $ P(x) $ satisfies the given conditions. By assumption, there exist polynomials $ Q(x) $ and $ R(x) $, of degree at most two, satisfying for all real values of the variable $ x $ the equations

The coefficient of $ x^2 $ in each of the polynomials $ Q(x) $, $ R(x) $ is equal to the coefficient of $ x^5 $ in the polynomial $ P(x) $ (we do not exclude the possibility that it is zero). The polynomials $ Q(x) $ and $ R(x) $ therefore have the form

Transforming the first of the above equations,

Transforming the second, we obtain, analogously,

Subtracting the obtained relations (the second from the first), we get a constant on the left side, equal to $ 2 $. Therefore, the polynomials on the right sides of the above relations must have equal coefficients for each of the positive powers of the variable $ x $, and their constant terms must differ by $ 2 $:

We thus have a system of $ 5 $ equations with $ 5 $ unknowns. From the second and fifth equations, we get the relationship

and from the third and fifth - 

Therefore, by adding and subtracting the sides,

From the first equation of the system, we have $ b - p = 6a $. Thus, $ 2a - 6 = 3(b - p) = 18a $, from which

Substituting these expressions into the fourth equation of the system, we determine $ c = -1 $. Therefore,

If the polynomial $ P(x) $ exists, then it must be given by each of the following two formulas:

Multiplying out, we obtain, from each of these formulas,

From the form of these formulas, it follows that the obtained polynomial indeed satisfies the conditions set in the problem.
Note 1. When solving the system of equations by elimination, we did not worry about writing an equivalent system of equations at each step. Therefore, at the end of the solution, it was necessary to verify that both proposed formulas indeed give the same polynomial $ P(x) $.