# Remainders with complex numbers

Let $f(x) \in C [x] .$
Suppose $f(-1+i) = 2+5i$ and $f(-2-i)=-3.$
Determine the remainder of f(x) divided by $(x+1-i)(x+2+i).$

How would i begin with this question, like how would i determine what f(x) is to begin with?

• Do you know how to approach this question if all if the coeffient of complex numbers were 0? Mar 29, 2014 at 5:21
• Yeah then its regular long division right? Mar 29, 2014 at 5:42

HINT:

Let $\displaystyle f(x)=A(x+1-i)(x+2+i)+B(x+2+i)+C(x+1-i)$

Put $x=-1+i$ and $-2-i$ one by one to find $B,C$

Alternatively,

let $\displaystyle f(x)=A(x+1-i)(x+2+i)+Bx+C$

Put $x=-1+i$ and $-2-i$ one by one to find $B,C$

In either cases, $B,C$ are arbitrary constants and $A$ is a polynomial

• What are $A,B,C$? Numbers? Polynomials? Mar 29, 2014 at 6:02
• @GerryMyerson, arbitrary constants independent of $x$ Mar 29, 2014 at 6:03
• So, you are assuming $f$ is a quadratic? Mar 29, 2014 at 6:03
• @GerryMyerson, sorry $B,C$ are arbitrary constants, $A$ is a polynomial Mar 29, 2014 at 6:05
• Maybe that information should be edited into your answer. Mar 29, 2014 at 6:06