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### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

# If $f\left(x\right),g\left(x\right)\in R\left[x\right]$ and $f\left(x\right)+g\left(x\right)\ne {0}_{R}$ , show that $deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{degf\left(x\right),deg\left[g\left(x\right)\right]\right\}$

It is proved that $deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{degf\left(x\right),deg\left[g\left(x\right)\right]\right\}$

See the step by step solution

## Step 1: Polynomial Arithmetic:

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

## Step 2: Polynomial Operations

Let the given functions be:

$f\left(x\right)=\sum _{i=0}^{m}{a}_{i}{x}^{i}.......\left\{m=degf\left(x\right)\right\}\phantom{\rule{0ex}{0ex}}g\left(x\right)=\sum _{j=0}^{m}{b}_{j}{x}^{j}.......\left\{n=degg\left(x\right)\right\}$

Now, assuming:

$i>m⇒{a}_{i}=0\phantom{\rule{0ex}{0ex}}j>n⇒{b}_{j}=0$

We have,

$\begin{array}{rcl}f\left(x\right)+g\left(x\right)& =& \left\{\sum _{i=0}^{m}{a}_{i}{x}^{i}\right\}+\left\{\sum _{j=0}^{m}{b}_{j}{x}^{j}\right\}\\ & =& \sum _{i=0}^{max\left(m,n\right)}\left({a}_{i}+{b}_{i}\right){x}^{i}\end{array}$

This implies that the degree of above expression will lie within:

$0\le i\le max\left(m,n\right).......\left\{{a}_{i}+{b}_{i}\ne 0\right\}$

.

This results: $deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{deg\left[f\left(x\right)\right],deg\left[g\left(x\right)\right]\right\}$

Hence proved, $deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{deg\left[f\left(x\right)\right],deg\left[g\left(x\right)\right]\right\}$.