Question

Using extended euclidean algorithm find f(x) and g(x) in: f(x)(x^5 + 4x^4 + 6x^3 + x^2 + 4x + 6) + g(x)(x^5 + 5x^4 + 10x^3 + x^2 + 5x + 10) = x^3+1

Answer 1

Lets denote

We have to find such that

Lets start with our division algorithm.

The last nonzero remainder is which can be written as linear combination of as follows:

(this is written based on the middle equation in the series of equations above)

(this is obtained by substituting the first equation in the series of equations above)

Simplifying, we get:

That is

So as to obtain the desired equation, dividing by 6:

Or

So we get   and