If a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2. Determine the remainder when the polynomial is divided by (x-1)(x+2).
We know that and where and are two polynomials such that .
Thus, and .
The remainer, when dividing by can be of degree 1 at most since is a polynomial of degree 2.
This is, and
where is a polynomial of degree
We see that and but we know that and so we've to solve a linear system of equations in the unknowns a and b. It's easy to solve this system and the solution is
Therefore, the reaminder when is divided by is