Polynomial long division

Start with a provisional polynomial quotient $q=0$ and remainder $h=f$. If $h$ has polynomial degree $n$ at least the divisor's polynomial degree $m$, with leading coefficients $a$ and $b$, subtract $tg$, whereThe identity $f=gq+h$ survives each step, while the leading term cancels and the polynomial degree of nonzero $h$ strictly decreases. Stop when $h=0$ or $\mathrm{deg}h<m$.

For uniqueness over a field, subtract two proposed decompositions:If $q_1\ne q_2$, the left side has polynomial degree at least $\mathrm{deg}g$, while the right side is zero or has smaller polynomial degree. Hence $q_1=q_2$ and then $r_1=r_2$.^[ref]