# GCD of $X^4+5X^3+3X^2+X$ and $X^2+1$ ( Euclidean Algorithm + Finding the inverse of the polynomial)

It is obvious the GCD of those two is 1.

The thing that makes me post this question is that why when using Euclidean Algorithm the GCD seems to be $\frac{5}{4}$ ?

I want furthermore to find the inverse of the polynomial and i know i can do that when the GCD = 1 by using Extended Euclidean Algorithm. Yet as i've said i have 5/4.

• $5/4$ is a unit (invertible) in the ring of polynomials with inverse $4/5$. It is like if you find that $-1$ is the gcd between two (rational) integers. People don't count an invertible factor.
– OR.
Jun 18, 2017 at 10:58
• Divide your whole equation by $5/4$ and you get an equation with 1.
– OR.
Jun 18, 2017 at 11:19
• OH RIGHT!! Thank you very much !! Jun 18, 2017 at 11:26
• That's the reason why people don't count units as interesting factors, you can divide them out as you need.
– OR.
Jun 18, 2017 at 11:29

So what? The conclusion (assuming that the computations are correct) is that $\frac54$ is a GCD of your polynomials. This is the same thing as saying that they are relatively prime.
• @Eduard6421 What's the problem? Start with $\frac54$ and express it as $\alpha(x)p(x)+\beta(x)q(x)$, where $p(x)$ and $q(x)$ are your polynomials. Then $1=\frac45\alpha(x)p(x)+\frac45\beta(x)q(x)$. Jun 18, 2017 at 11:18