How can you deduce a cubic polynomial given two quadratic divisors, their respective remainders, and nothing else?

Question

For example,

A cubic polynomial gives the remainders $(5x + 4)$ and $(12x - 1)$ when divided by $(x^2 - x + 2)$ and $(x^2 + x - 1)$ respectively.

How can I find the polynomial from this set of information?

Answer 1

Let $p(x)$ be the unknown polynomial.

Then for some unknown constants $a,b,c,d$, we must have the identities \begin{align*} p(x)&=(ax+b)(x^2-x+2)+(5x+4)\tag{1}\\[4pt] p(x)&=(cx+d)(x^2+x-1)+(12x-1)\tag{2}\\ \end{align*} Hence identically we have $$(ax+b)(x^2-x+2)+(5x+4)=(cx+d)(x^2+x-1)+(12x-1)$$ or equivalently, $$ax^3+(b-a)x^2+(2a-b+5)x+(2b+4)=cx^3+(c+d)x^2+(d-c+12)x-(d+1)$$ which yields the system \begin{align*} a&=c\\[4pt] b-a&=c+d\\[4pt] 2a-b+5&=d-c+12\\[4pt] 2b+4&=-d-1\\[4pt] \end{align*} of $4$ linear equations in $4$ unknowns.

To finish the task, solve the system and then substitute the results in either $(1)$ or $(2)$ to find $p(x)$.