Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 8.3, Problem 24E
To determine
To prove:
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Suppose that F(x) is a function such that F"(x) = f(x). Which of the following statements is true?
b
[ f(x) = F(b) – F(a)
[ f(x) = F(a) – F(b)
b
[* F(x) = f(b) – f(a)
b
* F(x) = f(a) – f(b)
-
a
Let f(x)=x2, g(x)=2x+5, and h(x)=x2-1
a. h[f(-2)]
b. f[g(x-1)]
Let f(x) = (x + 1)² -1, x2-1. Then the set
{x: f(x) = f(x)} is
Chapter 8 Solutions
Elements Of Modern Algebra
Ch. 8.1 - True or False
Label each of the following...Ch. 8.1 - Prob. 2TFECh. 8.1 - Prob. 3TFECh. 8.1 - Prob. 4TFECh. 8.1 - Prob. 5TFECh. 8.1 - Prob. 6TFECh. 8.1 - Prob. 7TFECh. 8.1 - Prob. 1ECh. 8.1 - Prob. 2ECh. 8.1 - Prob. 3E
Ch. 8.1 - Consider the following polynomial over Z9, where a...Ch. 8.1 - 5. Decide whether each of the following subset is...Ch. 8.1 - Determine which subset in Exercise 5 are ideals of...Ch. 8.1 - Prove that [ x ]={ a0+a1x+...+anxna0=2kfork }, the...Ch. 8.1 - Prob. 8ECh. 8.1 - Prob. 9ECh. 8.1 - Let R be a commutative ring with unity. Prove that...Ch. 8.1 - 11. a. List all the polynomials in that have...Ch. 8.1 - a. Find a nonconstant polynomial in Z4[ x ], if...Ch. 8.1 - Prob. 13ECh. 8.1 - 14. Prove or disprove that is a field if is a...Ch. 8.1 - 15. Prove that if is an ideal in a commutative...Ch. 8.1 - a. If R is a commutative ring with unity, show...Ch. 8.1 - Prob. 17ECh. 8.1 - 18. Let be a commutative ring with unity, and let...Ch. 8.1 - Prob. 19ECh. 8.1 - Consider the mapping :Z[ x ]Zk[ x ] defined by...Ch. 8.1 - Describe the kernel of epimorphism in Exercise...Ch. 8.1 - Assume that each of R and S is a commutative ring...Ch. 8.1 - Describe the kernel of epimorphism in Exercise...Ch. 8.1 - Prob. 24ECh. 8.1 - (See exercise 24.) Show that the relation...Ch. 8.2 - Label each of the following statements as either...Ch. 8.2 - Prob. 2TFECh. 8.2 - Prob. 3TFECh. 8.2 - Prob. 1ECh. 8.2 - Prob. 2ECh. 8.2 - Prob. 3ECh. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - Prob. 5ECh. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - Prob. 7ECh. 8.2 - Prob. 8ECh. 8.2 - Prob. 9ECh. 8.2 - Prob. 10ECh. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - Prob. 13ECh. 8.2 - Prob. 14ECh. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prob. 17ECh. 8.2 - Prob. 18ECh. 8.2 - Prob. 19ECh. 8.2 - Prob. 20ECh. 8.2 - Prob. 21ECh. 8.2 - Prob. 22ECh. 8.2 - Prob. 23ECh. 8.2 - Prob. 24ECh. 8.2 - Prob. 25ECh. 8.2 - Prob. 26ECh. 8.2 - Prob. 27ECh. 8.2 - Prob. 28ECh. 8.2 - Prob. 29ECh. 8.2 - Prob. 30ECh. 8.2 - Prob. 31ECh. 8.2 - Prob. 32ECh. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Prob. 35ECh. 8.3 - True or False
Label each of the following...Ch. 8.3 - Label each of the following statements as either...Ch. 8.3 - Prob. 3TFECh. 8.3 - True or False
Label each of the following...Ch. 8.3 - Prob. 5TFECh. 8.3 - Prob. 6TFECh. 8.3 - Prob. 7TFECh. 8.3 - True or False
Label each of the following...Ch. 8.3 - Prob. 9TFECh. 8.3 - Prob. 1ECh. 8.3 - Let Q denote the field of rational numbers, R the...Ch. 8.3 - Find all monic irreducible polynomials of degree 2...Ch. 8.3 - Write each of the following polynomials as a...Ch. 8.3 - Let F be a field and f(x)=a0+a1x+...+anxnF[x]....Ch. 8.3 - Prove Corollary 8.18: A polynomial of positive...Ch. 8.3 - Corollary requires that be a field. Show that...Ch. 8.3 - Let be an irreducible polynomial over a field ....Ch. 8.3 - Let be a field. Prove that if is a zero of then...Ch. 8.3 - Prob. 10ECh. 8.3 - Prob. 11ECh. 8.3 - Prob. 12ECh. 8.3 - Prob. 13ECh. 8.3 - Prob. 14ECh. 8.3 - Prob. 15ECh. 8.3 - Prob. 16ECh. 8.3 - Suppose that f(x),g(x), and h(x) are polynomials...Ch. 8.3 - Prove that a polynomial f(x) of positive degree n...Ch. 8.3 - Prove Theorem Suppose is an irreducible...Ch. 8.3 - Prove Theorem If and are relatively prime...Ch. 8.3 - Prove the Unique Factorization Theorem in ...Ch. 8.3 - Let ab in a field F. Show that x+a and x+b are...Ch. 8.3 - Let f(x),g(x),h(x)F[x] where f(x) and g(x) are...Ch. 8.3 - Prob. 24ECh. 8.3 - Prob. 25ECh. 8.3 - Prob. 26ECh. 8.3 - Prob. 27ECh. 8.4 - Label each of the following statements as either...Ch. 8.4 - Prob. 2TFECh. 8.4 - Prob. 3TFECh. 8.4 - Prob. 4TFECh. 8.4 - Prob. 5TFECh. 8.4 - Prob. 6TFECh. 8.4 - Prob. 7TFECh. 8.4 - Prob. 8TFECh. 8.4 - Prob. 9TFECh. 8.4 - Prob. 10TFECh. 8.4 - True or False
Label each of the following...Ch. 8.4 - Prob. 12TFECh. 8.4 - Prob. 13TFECh. 8.4 - Prob. 14TFECh. 8.4 - Prob. 15TFECh. 8.4 - 1. Find a monic polynomial of least degree over ...Ch. 8.4 - One of the zeros is given for each of the...Ch. 8.4 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - Prob. 5ECh. 8.4 - Prob. 6ECh. 8.4 - Prob. 7ECh. 8.4 - Prob. 8ECh. 8.4 - Prob. 9ECh. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 8.4 - Factor each of the polynomial in Exercise as a...Ch. 8.4 - Factor each of the polynomial in Exercise as a...Ch. 8.4 - Prob. 15ECh. 8.4 - Factors each of the polynomial in Exercise 1316 as...Ch. 8.4 - Prob. 17ECh. 8.4 - Show that the converse of Eisenstein’s...Ch. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Use Theorem to show that each of the following...Ch. 8.4 - Prob. 22ECh. 8.4 - Prove that for complex numbers .
Ch. 8.4 - Prob. 24ECh. 8.4 - Prob. 25ECh. 8.4 - Prob. 26ECh. 8.4 - Prob. 27ECh. 8.4 - Prob. 28ECh. 8.4 - Prob. 29ECh. 8.4 - Prob. 30ECh. 8.4 - Prob. 31ECh. 8.4 - Prob. 32ECh. 8.4 - Let where is a field and let . Prove that if is...Ch. 8.4 - Prob. 34ECh. 8.4 - Prob. 35ECh. 8.5 - Prob. 1TFECh. 8.5 - Prob. 2TFECh. 8.5 - Prob. 3TFECh. 8.5 - Prob. 4TFECh. 8.5 - Prob. 1ECh. 8.5 - Prob. 2ECh. 8.5 - Prob. 3ECh. 8.5 - Prob. 4ECh. 8.5 - Prob. 5ECh. 8.5 - Prob. 6ECh. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - Prob. 8ECh. 8.5 - Prob. 9ECh. 8.5 - Prob. 10ECh. 8.5 - Prob. 11ECh. 8.5 - Prob. 12ECh. 8.5 - Prob. 13ECh. 8.5 - Prob. 14ECh. 8.5 - Prob. 15ECh. 8.5 - Prob. 16ECh. 8.5 - Prob. 17ECh. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.5 - Prob. 20ECh. 8.5 - Prob. 21ECh. 8.5 - Prob. 22ECh. 8.5 - Prob. 23ECh. 8.5 - Prob. 24ECh. 8.5 - Prob. 25ECh. 8.5 - Prob. 26ECh. 8.5 - Prob. 27ECh. 8.5 - Prob. 28ECh. 8.5 - Prob. 29ECh. 8.5 - Prob. 30ECh. 8.5 - Derive the quadratic formula by using the change...Ch. 8.5 - Prob. 32ECh. 8.6 - True or False
Label each of the following...Ch. 8.6 - Prob. 2TFECh. 8.6 - Prob. 3TFECh. 8.6 - Prob. 1ECh. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 8.6 - In Exercises, a field , a polynomial over , and...Ch. 8.6 - In Exercises , a field , a polynomial over , and...Ch. 8.6 - In Exercises , a field , a polynomial over , and...Ch. 8.6 - Prob. 7ECh. 8.6 - If is a finite field with elements, and is a...Ch. 8.6 - Construct a field having the following number of...Ch. 8.6 - Prob. 10ECh. 8.6 - Prob. 11ECh. 8.6 - Prob. 12ECh. 8.6 - Prob. 13ECh. 8.6 - Prob. 14ECh. 8.6 - Prob. 15ECh. 8.6 - Each of the polynomials in Exercises is...Ch. 8.6 - Prob. 17ECh. 8.6 - Prob. 18E
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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?arrow_forwardLet f(x) = m1x + a and g(x) = m2x + b, where m1, m2, a, and b are constants.Determine the necessary and sufficient conditions on the constants so that f ∘ g = g ∘ f.arrow_forwardLet f(x) = 4.x³+ 2x- 10 and g(x) = x. (a) Prove that f(x) is O(g(x)). (b) Prove that g(x) is O(f(r)).arrow_forward
- Find functions g and h such that f(x) = g(h(x)). f(x) = 4x+2 A. g(x) = 4x+2, h(x) = ex O B. g(x)=ex, h(x) = 4x+2 C. g(x)= In x, h(x) = 4x+2 OD. g(x)=e4x, h(x)=x+2arrow_forwardIf f1(x) and f2(x) are functions from the set of positive integers to the set of positive real numbers and f1(x) and f2(x) are both Θ(g(x)), is ( f1 − f2)(x) also Θ(g(x))? Either prove that it is or give a counterexample.arrow_forwardThe functions f(x) and g(x) are continuous on the interval (a,b). Determine which of the following are continuous and which are discontinuous. g(x) + 9(x – 1)* 9 f(x) f(x) • 9(x- 1)* f(g(x)) Ig(x)| f(x) + g(x) f(x)•g(x) f(x) + g(x) g(x) f(x) 9(f(x) + g(x))* 9(x- 1)ª 9(x - 1)* g(x) Drag each of the above items into the appropriate area below, depending on whether it represents a continuous or discontinuous function. Continuous Discontinuousarrow_forward
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