1. ## Problem #18: Polynomial

Suppose p(x) is a polynomial of degree 7 such that $(x - 1)^4$ is a factor of p(x) + 1 and $(x + 1)^4$ is a factor of p(x) - 1. Find p(x).

(Source: Trinity University Mathematics POTW)

-Dan

2. ## Re: Problem #18: Polynomial

Hint:

Spoiler:

$p(x) = (x - 1)^4 \cdot Q(x)$ where Q(x) is a polynomial of degree 3.

What can you say about the derivative of p(x)?

-Dan

3. ## Re: Problem #18: Polynomial

To get you started, write the formulas for what the two statements mean.
p(x) + 1 = ...
P(x) - 1 = ...

4. ## Re: Problem #18: Polynomial

Dan,
Here's a solution assuming the polynomial is over the reals (or the complex field); by the way, good hint.
Spoiler:

Good job!

-Dan

6. ## Re: Problem #18: Polynomial

Thanks and congratulations to Ishuda and particularly johng for their efforts. No need to post my solution this week as johng's post is virtually identical to mine.

-Dan