A Book of Abstract Algebra

Charles C. Pinter

Chapter 24 RINGS OF POLYNOMIALS - all with Video Answers

Educators

Section 1

Problem 1

Let $a(x)=2 x^{2}+3 x+1$ and $b(x)=x^{3}+5 x^{2}+x .$ Compute $a(x)+b(x)$ $a(x)-b(x)$ and $a(x) b(x)$ in $\mathbb{Z}[x], \mathbb{Z}_{5}[x], \mathbb{Z}_{6}[x]$, and $\mathbb{Z}_{7}[x]$

Carole Wastog

Numerade Educator

02:22

Problem 2

Find the quotient and remainder when $x^{3}+x^{2}+x+1$ is divided by $x^{2}+3 x+2$ in $\mathbb{Z}[x]$ and in $\mathbb{Z}_{5}[x] .$

Ankit Gupta

Numerade Educator

02:20

Problem 3

Find the quotient and remainder when $x^{3}+2$ is divided by $2 x^{2}+3 x+4$ in $\mathbb{Z}[x]$ in $\mathbb{Z}_{3}[x]$, and in $\mathbb{Z}_{5}[x]$

We call $b(x)$ a factor of $a(x)$ if $a(x)=b(x) q(x)$ for some $q(x)$, that is, if the remainder when $a(x)$ is divided by $b(x)$ is equal to zero.

Saurabh Chandra

Numerade Educator

01:24

Problem 4

Show that the following is true in $A[x]$ for any ring $A:$ For any odd $n$,
(a) $x+1$ is a factor of $x^{n}+1$
(b) $x+1$ is a factor of $x^{n}+x^{n-1}+\cdots+x+1$

Tanishq Gupta

Numerade Educator

08:12

Problem 5

Prove the following: In $\mathbb{Z}_{3}[x], x+2$ is a factor of $x^{m}+2$, for all $m .$ In $\mathbb{Z}_{n}[x]$, $x+(n-1)$ is a factor of $x^{m}+(n-1)$, for all $m$ and $n$.

Bobby Barnes

University of North Texas

02:14

Problem 6

Prove that there is no integer $m$ such that $3 x^{2}+4 x+m$ is a factor of $6 x^{4}+50$ in $\mathbb{Z}[x]$

Teresa Fuston

Numerade Educator