Question

what must be added to f(x)=x^4+2x^3-2x^2+x -1 so that the resulting polynomial is exactly divisible by g(x)=x^2+2x-3

Answer 1

Let's pretend to divide:

Answer 2

\[x^4+2x^3-2x^2+x -1\div x^2+2x-3=x^2+1, remainder -x\]

Answer 3

That remainder is what we want to remove, so add the opposite in. Then test the division again to be sure you have it right.

Answer 4

alternatively, you could have factored x^2+2x -3, found out its roots and then tried to have satisfied it in x^4 + 2x^3 - 2x^2 + x -1 + Ax + B =0