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Abstract Algebra: An Introduction
Found in: Page 94
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

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

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Short Answer

If fx,gxRx and fx+gx0R , show that


It is proved that degfx+gxmaxdegfx,deggx

See the step by step solution

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:

fx=i=0maixi .......m=deg fxgx=j=0mbjxj .......n=deg gx

Now, assuming:


We have,

fx+gx=i=0maixi+j=0mbjxj =i=0maxm,nai+bixi

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

0imaxm,n .......ai+bi0


This results: degfx+gxmaxdegfx, deggx

Hence proved, degfx+gxmaxdegfx, deggx.

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