they can be computed by . We denote the right quotient by $Q = \operatorname{{rquo}}(F, G)$ and the right remainder by $R = \operatorname{{rrem}}(F, G)$. The latter is used in (Euclidean algorithm) for computing the left gcd of two quaternion polynomials $F, G \in {\mathbb{H}}[t]$ in case of monic $G$. The left gcd is the unique monic polynomial $L = \operatorname{{lgcd}}(F,G)$ of maximal degree such that there exist polynomials $Q, R \in {\mathbb{H}}[t]$ with $F = LQ$ and $G = LR$. The function $\operatorname{{lcoeff}}$ in Line \[line:lcoeff\] of returns the leading coefficient of a polynomial so that $\operatorname{{lcoeff}}(R)^{-1}R$ is monic.

Two polynomials $F, G \in {\mathbb{H}}[t]$, $G$ is monic. Polynomials $Q, R \in {\mathbb{H}}[t]$ such that $\deg R < \deg G$ and $F = GQ + R$. $Q \leftarrow 0$, $R \leftarrow F$ $m \leftarrow \deg F$, $n \leftarrow \deg G$ $c \leftarrow \operatorname{{lcoeff}}(R)$ leading coefficient of $R$ $Q \leftarrow Q + ct^{m-n}$ \[line:Gct\]$R \leftarrow R - Gct^{m-n}$ $m \leftarrow \deg R$ $Q, R$

Two polynomials $F$, $G \in {\mathbb{H}}[t]$, $G$ is monic. Monic polynomial $L \in {\mathbb{H}}[t]$ such that there exist polynomial $Q, R \in {\mathbb{H}}[t]$ with $F = LQ$ and $G = LR$. $R \leftarrow \operatorname{{rrem}}(F, G)$ $G$ 