The division algorithm has an analoguen in $\displaystyle Q(x)$ that asserts: given two polynomials $\displaystyle f(x),g(x),g(x)$ not identically 0, there exists $\displaystyle q(x),r(x)\in{Q}(x)$ such that

$\displaystyle f(x)=g(x)q(x)+r(x)$

and either

$\displaystyle r(x)\equiv0$, or

$\displaystyle 0\leq\text{degree of }r(x)<\text{degree of }g(x)$.

Prove this. One route is to proceed by the following steps:

a) If $\displaystyle f(x)\equiv0$ or the degree of $\displaystyle f(x)$ is 0, prove the assertion.

b) Proceed by induction on the degree of $\displaystyle f(x)$ and assume results for all cases where $\displaystyle f(x)$ is of degree less than $\displaystyle n$. Let $\displaystyle f(x)$ be of degree $\displaystyle n$.

Subcase (1) $\displaystyle \text{degree }g(x)>\text{degree }f(x)$

Prove directly

Subcase (2) $\displaystyle \text{degree }f(x)\geq\text{degree }g(x)$

Form polynomial $\displaystyle h(x)=f(x)-(\frac{a_n}{b_n})x^{n-m}g(x)$ and use inductive hypothesis.

OK. If someone could just do the first part of this, a), then I think that MAYBE I can do the rest. Please don't just go and post an entire solution. I really want to try and do some of this. I just can't get started.

Thanks to anyone willing to walk me through this.

Note: The polynomial $\displaystyle h(x)$ was discussed in the previous problem here ---> http://www.mathhelpforum.com/math-he...hm-170136.html