Let $(x - 1)^3$ divide $(P(x) + 1)$ and $(x + 1)^3$ divide$ (P(x) - 1)$. Find the polynomial $P(x)$ of degree $5$.

Question

I tried utilising the divisibility laws but to no result. I then tried bujte force to find two polynomials of degree 2 that satisfy both the equations. But still got no result. Please help me.

However, i don't want to use chinese remainder theorem i want to get an answer using plain old algebra — SAGNIK UPADHYAY, Sep 03 '20 at 17:58
https://math.stackexchange.com/questions/1740767/find-a-polynomial-fx-of-degree-5-such-that-2-properties-hold/2476852#2476852 — nonuser, Sep 03 '20 at 18:04

score 1 · Answer 1 · answered Sep 03 '20 at 17:59

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Hint: What can you say about the factors of $P'(x)$?

answered Sep 03 '20 at 17:59

Arthur

199,419

That's a good hint :-) – Jyrki Lahtonen Sep 03 '20 at 18:00
Um should I delete the question, since it got downvoted – SAGNIK UPADHYAY Sep 03 '20 at 18:01
But I still did not understand – SAGNIK UPADHYAY Sep 03 '20 at 18:01
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@JyrkiLahtonen The hint/method is also a dupe - which of course is not the least but surprising since questions like this are FAQs (so one should resist the temptation to add/encourage yet more dupe answers) – Bill Dubuque Sep 03 '20 at 18:28
A good catch @BillDubuque. This time I didn't have the time to look for a duplicate - apparently neither did Arthur. – Jyrki Lahtonen Sep 03 '20 at 18:37

Let $(x - 1)^3$ divide $(P(x) + 1)$ and $(x + 1)^3$ divide$ (P(x) - 1)$. Find the polynomial $P(x)$ of degree $5$.

1 Answers1