<s>[INST] Determine polynomial of $3$ degree real coefficients $f(x)$ such that $f(x) \vdots (x-2)$ and $f(x)$ dividing by $x^2-1$ have remainder $2x$ I need help in this problem.

Problem: Determine polynomial of $3$ degree real coefficients $f(x)$ such that $f(x) \vdots (x-2)$ and $f(x)$ divided by $x^2-1$ has remainder $2x$ and suppose $f(x)=x^3+ax^2+bx+c$.

 [/INST] Let $f(x) = ax^3 + bx^2 + cx + d$
and also $f(x) = (px+q)(x^2-1) + 2x$
Comparing the coefficients of $f(x)$, we have
$a = p$, $b = q$, $c = 2 - p$, $d = -q$
Also $f(2) = 0 \implies 8a + 4b + 2c + d = 0$
Substituting the values of $a, b, c$ in this equation and simplifying
$6p + 3q + 4 = 0$
You can find infinite number of solutions to the above equation.
For example, if you choose $p = 1$ then $q = -\frac{10}{3}$
and $f(x) = x^3 -\frac{10}{3}x^2 + x + \frac{10}{3}$
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