I'm having trouble solving this problem in my text book and I'm just not sure where to start/what the hint is trying to tell me.
Let $p(x)$ be an irreducible polynomial of degree $m$, and let $r(x)$ be a polynomial of degree $<mv$. Show that there are polynomials $r_1(x),r_2(x),…,r_v(x)$ with $\deg(r_j(x))<m$ so that: $$\frac{r(x)}{p(x)^v}=\frac{r_1(x)}{p(x)}+\frac{r_2(x)}{p(x)^2}+\frac{r_3(x)}{p(x)^3}+\cdots+\frac{r_v(x)}{p(x)^v}.$$
Hint: $r_1(x)$ is the quotient when $r(x)$ is divided by $p(x)^{v-1}$.
Best Answer
Multiply both sides by $p(x)^v$. We get
$$r(x)=r_1(x)p(x)^{v-1}+r_2(x)p(x)^{v-2}+\ldots+r_{v-1}(x)p(x)^{1}+r_v(x).$$
Since $\text{deg}(r_v)<\text{deg}(p)$ we get that $r_v$ is the remainder of the division of $r(x)$ by $p(x)$. Notice that
$$\frac{r(x)-r_v(x)}{p(x)^v}=r_1(x)p(x)^{v-2}+r_2(x)p(x)^{v-3}+\ldots+r_{v-1}(x)$$
is the quotient. So, $r_{v-1}$, $r_{v-2}$, ... can be found in a similar way, by computing the remainder by division by $p(x)$ of this quotient.