Question

without actual division,prove that a4 +2a3-2a2+2a-3 is exactly divisible by a2 +2a-3

Answer 1

Let f = $a^4 + 2a^3-2a^2+2a-3$

The roots of $a^2+2a-3$ are -3,1. (or (x+3)(x-1) are factors of this equation)

By remainder theorem (x-a) is a factor of a function f, iff f(a) = 0
We need to show that (x+3) and (x-1) are also factors of $a^4 + 2a^3-2a^2+2a-3$. This can be shown by the above discussed remainder theorem.

It can be checked that f(1) = 0 and f(-3) = 0, Hence the $a^2+2a-3$ is a factor of $a^4 + 2a^3-2a^2+2a-3$.