Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $\bullet$ $f(x)$ is divisible by $x^3$. $\bullet$ $f(x) 2$ is divisible by $(x 1)^3$. Write your answer in expanded form (that is, do not factor $f(x)$).

There seems to be one character missing. But I gather that f(x) needs to satisfy

•  divides

•  divides

I'll also assume f(x) is monic, meaning the coefficient of the leading term is 1, or

Since divides , and

where is degree-2, and we can write it as

Now, we have

so if divides , then is degree-2, so is degree-4, and we can write

where is degree-1.

Expanding the left side gives

and dividing by leaves no remainder. If we actually compute the quotient, we wind up with

If the remainder is supposed to be zero, then

Adding these equations together and grouping terms, we get

Then , and you can solve for a and b by substituting this into any of the three equations above. For instance,

So, we end up with