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Abstract Algebra

John A. Beachy; William D. Blair

Chapter 4

Polynomials - all with Video Answers

Educators

Section 1

Fields; Roots of Polynomials

01:44

Problem 1

Let $f(x), g(x), h(x) \in F[x]$. Show that the following properties hold.
(a) If $g(x) \mid f(x)$ and $h(x) \mid g(x),$ then $h(x) \mid f(x)$.
(b) If $h(x) \mid f(x)$ and $h(x) \mid g(x),$ then $h(x) \mid(f(x) \pm g(x))$.
(c) If $g(x) \mid f(x),$ then $g(x) \cdot h(x) \mid f(x) \cdot h(x)$.
(d) If $g(x) \mid f(x)$ and $f(x) \mid g(x),$ then $f(x)=k g(x)$ for some $k \in F$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:01

Problem 2

Let $p$ be a prime number, and let $n$ be a positive integer. How many polynomials are there of degree $n$ over $\mathbf{Z}_{p}$ ?

Carson Merrill
Carson Merrill
Numerade Educator
01:27

Problem 3

For $f(x)=2 x^{3}+x^{2}-2 x+1,$ use the method of Theorem 4.1 .9 to write $f(x)=$ $q(x)(x-1)+f(1)$

Taylor Shimono
Taylor Shimono
Numerade Educator
01:36

Problem 4

For $f(x)=x^{3}+3 x^{2}-10 x+5,$ use the method of Theorem 4.1 .9 to write $f(x)=$ $q(x)(x-2)+f(2)$

Jessica Kluck
Jessica Kluck
Numerade Educator
02:27

Problem 5

Over the given field $F$, write $f(x)=q(x)(x-c)+f(c)$ for
(a) $f(x)=2 x^{3}+x^{2}-4 x+3 ; \quad c=1 ; \quad F=\mathbf{Q}$
(b) $f(x)=x^{3}-5 x^{2}+6 x+5 ; \quad c=2 ; \quad F=\mathbf{Q}$
(c) $f(x)=x^{3}+1 ; \quad c=1 ; \quad F=\mathbf{Z}_{3}$
(d) $f(x)=x^{3}+2 x+3 ; \quad c=2 ; \quad F=\mathbf{Z}_{5}$.

Gregory Higby
Gregory Higby
Numerade Educator
01:59

Problem 6

Let $p$ be a prime number. Find all roots of $x^{p-1}-1$ in $\mathbf{Z}_{p}$.

James Chok
James Chok
Numerade Educator
01:18

Problem 7

Prove that if $p$ is a prime number, then the multiplicative group $\mathbf{Z}_{p}^{\times}$ is cyclic.

James Chok
James Chok
Numerade Educator
00:57

Problem 8

Let $p$ be a prime number, and let $a, b \in \mathbf{Z}_{p}^{\times}$. Show that if neither $a$ nor $b$ is a square, then $a b$ is a square.

Trang Hoang
Trang Hoang
Numerade Educator
01:34

Problem 9

Show that if $c$ is any element of the field $F$ and $k>2$ is an odd integer, then $x+c$ is a factor of $x^{k}+c^{k}$.

AG
Ankit Gupta
Numerade Educator
01:17

Problem 10

Show that rational numbers correspond to decimals which are either repeating or terminating. Hint: If $q=m / n,$ then when dividing $m$ by $n$ to put $q$ into decimal form there are at most $n$ different remainders. Conversely, if $d$ is a repeating decimal, then find $s, t$ such that $10^{s} d-10^{t} d$ is an integer.

Carson Merrill
Carson Merrill
Numerade Educator
02:03

Problem 11

Let $a$ be a nonzero element of a field $F$. Show that $\left(a^{-1}\right)^{-1}=a$ and $(-a)^{-1}=-a^{-1}$.

James Chok
James Chok
Numerade Educator
03:42

Problem 12

Let $a, b, c$ be elements of a field $F$. Prove that if $a \neq 0,$ then the equation $a x+b=c$ has a unique solution.

Aman Gupta
Aman Gupta
Numerade Educator
00:28

Problem 13

Show that the set $\mathbf{Q}(\sqrt{3})=\{a+b \sqrt{3} \mid a, b \in \mathbf{Q}\}$ is closed under addition, subtraction, multiplication, and division.

Amy Jiang
Amy Jiang
Numerade Educator
05:50

Problem 14

Let $F$ be any field. An $n \times n$ matrix with entries in $F$ is called a scalar matrix if it has the form $a I,$ where $I$ is the $n \times n$ identity matrix, and $a \in F$. Prove that the set of all $n \times n$ scalar matrices over $F$ is a field under the operations of matrix addition and multiplication.

Wendy Wang
Wendy Wang
Numerade Educator
01:55

Problem 15

Show that the set of matrices of the form $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right],$ where $a, b \in \mathbf{R},$ is a field under the operations of matrix addition and multiplication.

Thomas Emment
Thomas Emment
Numerade Educator
01:55

Problem 16

Show that the set of matrices of the form $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right],$ where $a, b \in \mathbf{R},$ is a field under the operations of matrix addition and multiplication.

Thomas Emment
Thomas Emment
Numerade Educator
03:51

Problem 17

Complete the proof that the set of matrices in Example 4.1 .3 is a field.

Wasim Sher
Wasim Sher
Numerade Educator
01:08

Problem 18

Show that the following set of matrices over $\mathbf{Z}_{2}$ is a field under the operations of matrix addition and multiplication:
$\begin{array}{l}{\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right],} & {\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right],} & {\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0\end{array}\right],} & {\left[\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1\end{array}\right],} \\ {\left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right],} & {\left[\begin{array}{lll}1 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right],} & {\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1\end{array}\right],} & {\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 0\end{array}\right] .}\end{array}$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:47

Problem 19

Let $\left(x_{0}, y_{0}\right),\left(x_{1}, y_{1}\right),$ and $\left(x_{2}, y_{2}\right)$ be points in the Euclidean plane $\mathbf{R}^{2}$ such that $x_{0}, x_{1}, x_{2}$ are distinct. Show that the formula
$$
f(x)=\frac{y_{0}\left(x-x_{1}\right)\left(x-x_{2}\right)}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}+\frac{y_{1}\left(x-x_{0}\right)\left(x-x_{2}\right)}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)}+\frac{y_{2}\left(x-x_{0}\right)\left(x-x_{1}\right)}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)}
$$
defines a polynomial $f(x)$ such that $f\left(x_{0}\right)=y_{0}, f\left(x_{1}\right)=y_{1},$ and $f\left(x_{2}\right)=y_{2}$. Note: This is a special case of Lagrange's interpolation formula.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
01:55

Problem 20

Use Lagrange's interpolation formula to find a polynomial $f(x)$ such that $f(1)=0$, $f(2)=1,$ and $f(3)=4$

Manisha Sarker
Manisha Sarker
Numerade Educator
02:19

Problem 21

Find a polynomial $f(x)$ such that $f(1)=-15, f(0)=3, f(2)=-3,$ and $f(4)=15$.

Devon Kelley
Devon Kelley
Numerade Educator
08:41

Problem 22

Show that $f(x)=b_{1}+\frac{1}{2}\left(b_{2}-b_{0}\right)\left(\frac{x-a}{h}\right)+\frac{1}{2}\left(b_{2}-2 b_{1}+b_{0}\right)\left(\frac{x-a}{h}\right)^{2}$ has the property that $f(a-h)=b_{0}, f(a)=b_{1},$ and $f(a+h)=b_{2}$. Note: Exercises 22 and 23 provide the basis for Simpson's rule for numerical integration via parabolic approximations.

Kevin Harmer
Kevin Harmer
Numerade Educator
00:51

Problem 23

For the polynomial $f(x)$ in Exercise $22,$ find $\int_{a-h}^{a+h} f(x) d x$.

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:54

Problem 24

Is it possible to define a quadratic polynomial whose graph contains the four points $(-1,-2),(0,-2),(1,0),$ and (2,2)$?$

Sara Sasani
Sara Sasani
Numerade Educator
05:04

Problem 25

(a) Extend the formula of Exercise 19 to the case of four points in the plane.
(b) Extend the formula of Exercise 19 to the case of $k$ points in the plane.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
04:40

Problem 26

From Exercises 2 and 3 of Section A.4 of the appendix, it appears to be a reasonable conjecture that $\sum_{i=1}^{n} i^{k}$ is a polynomial of degree $k+1$ in $n,$ e.g. $\sum_{i=1}^{n} i=$ $\left(n^{2}+n\right) / 2$ and $\sum_{i=1}^{n} i^{2}=\left(2 n^{3}+3 n^{2}+n\right) / 6$. We will assume, for the moment, that $\sum_{i=1}^{n} i^{k}=P_{k+1}(n)$ is a polynomial in $n$ of degree $k+1,$ and then attempt to find $P_{k+1}(n)$ by using the formula from Exercise $25 .$ For this purpose we need $k+2$ points that the polynomial passes through. We get these by evaluating the sums $\sum_{i=1}^{n} i^{k}=P_{k+1}(n)$ for $n=1,2, \ldots, k+2$. (Any $k+2$ values of $n$ will do, but these are easy to compute.)

Sirat Shah
Sirat Shah
Numerade Educator