<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] Because $f$ is divisible by $x^3$, we must have $f(x)=r_3x^3+r_4x^4+r_5x^5$.
The second condition gives, comparing coefficients of powers of $x$, that
$$
f(x)=6x^5 - 15x^4 + 10x^3.
$$
Edit: Comparing coefficients gives me the linear equations $r_4 + 3r_5 - 3=0, r_3 - 3r_5 + 8$ and $r_5=6$.
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