Algebra Polynomial Help

Find (without calculus) a fifth degree polynomial p(x) such that p(x) + 1 is divisible by (x - 1)^3 and p(x) - 1 is divisible by (x + 1)^3.

In one solution I found, the following properties were used (given without proof) but I am not sure why those properties are so obvious.

1) If p(x) + 1 is divisible by (x-1)^3, then p(-x) -1 is also divisible by (x-1)^3.
2) If p(x) - 1 is divisible by (x + 1)^3 then p(-x) + 1 is also divisible by (x+1)^3.

Since $p(x) + 1 = k(x-1)^3 \Rightarrow p(x) = k(x-1)^3 - 1$ then $p(-x) - 1 = k(-x-1)^3 + 1 -1 = -k(x+1)^3$, etc...

Thank you for the reply. As p(x) is a fifth degree polynomial, k should be a second degree polynomial not a constant. If you replace x by -x, k no longer remains same.

Would you mind telling me flaw(if there is) in above argument?

A shame you can't use calculus. Makes it really really easy.
Anywho, the standard writing p(x)=ax^5+ etc
Solve for you coefficients the -x makes sure when solving simultaneous eqs adding will cancel some coefficients and recall values of p(1) and p(-1) and a few others and you should be done.


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It is British Mathematics Olympiad question and it is asked to solve without Calculus. I am comfortable with Calculus.

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What year?
I did all the questions from late 90s to now.


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