= Quotient of polynomial division
{title2=$s$}
{title2=$q$}
= Polynomial quotient
{synonym}
In <polynomial division> $q=Qs+r$ over a <field>, the polynomial quotient is the <polynomial> $s$. The <polynomial remainder> $r$ has smaller <polynomial degree> than the nonzero divisor $Q$ unless it is zero.
In the notation $f=gq+r$, the <polynomial quotient> is instead named $q$. It is a <polynomial> determined by the divisor and the <polynomial degree> bound. The pointwise ratio of the associated <polynomial functions> satisfies
$$
\frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}\qquad\text{when }g(x)\ne0.
$$
For the worked example, $f(0)/g(0)=6/(-2)=-3$, while $q(0)=1$; the correction is $r(0)/g(0)=8/(-2)=-4$. Thus a <polynomial quotient> need not equal the <rational function> $f/g$. At $x=1$ and $x=-2$, the pointwise ratio is undefined, but the <polynomial quotient> still has perfectly well-defined values. When $r=0$, the divisor is a <polynomial factor> of $f$ and the ratio agrees with $q$ wherever $g(x)\ne0$.
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