$(x+b)^3\mid P(x)+a$ and $(x-a)^3\mid P(x)-a$

Question

$a,b\in\mathbb{C}$, $b!=0$

I need to find all the polynomials $P$ of degree $5$ verifying:

$ \begin{cases}(x+b)^3\mid P(x)+a\\ (x-b)^3\mid P(x)-a\end{cases} $ PS : there was en error, i fixed it

How would I do that?

Answer 1

Hint: $(x+b)^3|P(x)+a$ means $P(x)+a = (x+b)^3Q(x)$, with $Q$ quadratic.

Therefore

$$P(x)-a= (x+b)^3Q(x)-2a \,.$$

Now recall that for a Polynomial $H(x)$ you have $(x-a)^3|H(x)$ if and only if $H(a)=H'(a)=H''(a)=0$.

If you apply this to $H(x)=(x+b)^3Q(x)-2a$, you will find $Q(a), Q'(a)$ and $Q''(a)$. This is enough to find the Quadratic polynomial $Q(x)$.