<s>[INST] Find a polynomial f(x) of degree 5 such that 2 properties hold. I have been trying to find a polynomial $f(x)$ such that these $2$ properties hold:


*

*$f(x)-1$ is divisible by $(x-1)^3$

*$f(x)$ is divisible by $x^3$
To start, I set $f(x) =ax^5 + bx^4 + cx^3 + dx^2 + ex + f$.
This is divisible by $x^3$, so $d, e, f $  must be $0$. So polynomial $f(x) = ax^5 + bx^4 + cx^3$
This minus $1$ is divisible by $(x-1)^3$. So I used synthetic division and got that the remainder of $ax^5 + bx^4 + cx^3$ divided by $(x-1)$ is $a+b+c-1$. But it should divide out evenly, so $a+b+c-1 = 0$.
From here, I can't find any other equations involving the three variables. Could someone have any suggestions for how to continue? Thanks in advance. :)
 [/INST] Since $x^2$ and $(x-1)^2$ divide $p'(x)$ we have $$p'(x) =ax^2(x-1)^2 =ax^4-2ax^3+ax^2,$$ thus $$p(x) = {a\over 5}x^5-{a\over 2}x^4+{a\over 3}x^3+c$$
Since $p(0)=0$ we have $c=0$ and since $p(1)=1$ we have ${a\over 5}-{a\over 2}+{a\over 3}=1$ so $a=30$.
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