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8th Edition

Gilbert + 2 others

Publisher: Cengage Learning,

ISBN: 9781285463230

Chapter 8.4, Problem 27E

Textbook Problem

Derive the quadratic formula for the zeros of

Expert Solution

To determine

The quadratic formula for the zeros of

Let

To find zeros of

Consider,

Subtract

As

By adding

Elements Of Modern Algebra

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Ch. 8.1 - True or False
Label each of the following...Ch. 8.1 - True or False
Label each of the following...Ch. 8.1 - Label each of the following statements as either...Ch. 8.1 - True or False
Label each of the following...Ch. 8.1 - Label each of the following statements as either...Ch. 8.1 - Label each of the following statements as either...Ch. 8.1 - True of False
Label each of the following...Ch. 8.1 - Write the following polynomials in expanded form....Ch. 8.1 - 2. Express the following polynomials by using...Ch. 8.1 - 3. Consider the following polynomials over , where...

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 - 8. a. Prove that
,
the set of all polynomials with...Ch. 8.1 - 9. a. Let be a field. Prove that
,
the set of all...Ch. 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 - 13. a. How many polynomials of degree 2 are there...Ch. 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 - 17. a. Suppose that is a commutative ring with...Ch. 8.1 - 18. Let be a commutative ring with unity, and let...Ch. 8.1 - In the integral domain Z[ x ], let (Z[ x ])+...Ch. 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 - For each f(x)=i=0naixi in R[x], the formal...Ch. 8.1 - (See exercise 24.) Show that the relation...Ch. 8.2 - Label each of the following statements as either...Ch. 8.2 - True or False
Label each of the following...Ch. 8.2 - True or False
Label each of the following...Ch. 8.2 - For,, and given in Exercises , find and in that...Ch. 8.2 - For and given in Exercises , find and in that...Ch. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 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 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For , , and given in Exercises 7-10, find the...Ch. 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 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For , , and given in Exercises , find and in ...Ch. 8.2 - a. Factor x as a product of two polynomials of...Ch. 8.2 - Factor each of the following polynomials as the...Ch. 8.2 - 17. Factor each of the following polynomials as...Ch. 8.2 - Prove or disprove that the polynomial can be...Ch. 8.2 - Let I be the principal ideal (x2+1)={...Ch. 8.2 - 20. Let be the principal ideal . Determine...Ch. 8.2 - 21. Let be the principal ideal . Determine...Ch. 8.2 - 22. Let be the principal ideal . Determine...Ch. 8.2 - 23. Let where . Prove .
Ch. 8.2 - Let f(x)=anxn+an1xn1++a0 where an0. Find the...Ch. 8.2 - 25. Prove that if and are nonzero elements of ...Ch. 8.2 - Prove that if d1(x) and d2(x) are monic...Ch. 8.2 - 27. Show that the polynomials and in the...Ch. 8.2 - Prove that if h(x)|f(x) and h(x)|g(x) in F[ x ],...Ch. 8.2 - 29. Let . Prove that if and then .
Ch. 8.2 - 30.In the statement of the Division...Ch. 8.2 - 31. With the notation used in the description of...Ch. 8.2 - Prove that every nonzero remainder rj(x) in the...Ch. 8.2 - Prove that the only elements of F[ x ] that have...Ch. 8.2 - Prove that every ideal in where is a field, is a...Ch. 8.2 - Follow the pattern in Definition of Section to...Ch. 8.3 - True or False
Label each of the following...Ch. 8.3 - Label each of the following statements as either...Ch. 8.3 - Label each of the following statements as either...Ch. 8.3 - True or False
Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - True or False
Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - True or False
Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - 1. Determine the remainder when is divided by ...Ch. 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 - Let and be two polynomials over the field , both...Ch. 8.3 - Let be a prime integer, and consider the...Ch. 8.3 - Find all the zeros of each of the following...Ch. 8.3 - Give an example of a polynomial of a degree 4 over...Ch. 8.3 - If and are polynomials over the field , and ,...Ch. 8.3 - If f(x) and g(x) are polynomials over the field F,...Ch. 8.3 - Let f(x) be a polynomial of positive degree n over...Ch. 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 - Let where and are relatively prime. If and ,...Ch. 8.3 - Let f(x),g(x),h(x)F[x] where f(x) and g(x) are...Ch. 8.3 - Let and the greatest common divisor of and ...Ch. 8.3 - Find the least common multiple of each pair of...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False
Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False
Label each of the following...Ch. 8.4 - True or False
Label each of the following...Ch. 8.4 - True or False
Label each of the following...Ch. 8.4 - True or False
Label each of the following...Ch. 8.4 - True or False
Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 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 - Find all rational zeros of each of the polynomial...Ch. 8.4 - Find all rational zeros of each of the polynomial...Ch. 8.4 - Find all rational zeros of each of the polynomial...Ch. 8.4 - Find all rational zeros of each of the polynomial...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise 712, Find all zeros of the given...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise 712, Find all zeros of the given...Ch. 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 - Factor each of the polynomial in Exercise as a...Ch. 8.4 - Factors each of the polynomial in Exercise 1316 as...Ch. 8.4 - Show that each of the following polynomials is...Ch. 8.4 - Show that the converse of Eisenstein’s...Ch. 8.4 - Let be a polynomial of positive degree with...Ch. 8.4 - Show that each of the following polynomials is...Ch. 8.4 - Use Theorem to show that each of the following...Ch. 8.4 - Show that converse of Theorem is not true by...Ch. 8.4 - Prove that for complex numbers .
Ch. 8.4 - Prove that z1z2zn=z1z2zn for complex numbers...Ch. 8.4 -
Let be prime. Use Eisenstien’s Criterion to...Ch. 8.4 - Let f(x)=a0+a1x++an1xn1+xn be a monic polynomial...Ch. 8.4 - Derive the quadratic formula for the zeros of...Ch. 8.4 - Prove Theorem . (Hint: In the factorization...Ch. 8.4 - Prove that any polynomial of odd degree that has...Ch. 8.4 - Let in . Prove that if for all or if for all,...Ch. 8.4 - Let in . Prove that if the coefficients ...Ch. 8.4 - Let be in the field. Define the mapping by ....Ch. 8.4 - Let where is a field and let . Prove that if is...Ch. 8.4 - Show that each of the following polynomials is...Ch. 8.4 - Prove that is irreducible over for any prime....Ch. 8.5 - True or False
Label each of the following...Ch. 8.5 - True or False
Label each of the following...Ch. 8.5 - True or False
Label each of the following...Ch. 8.5 - True or False
Label each of the following...Ch. 8.5 - In Exercises 118, use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises 118, use the techniques presented in...Ch. 8.5 - In Exercises 118, use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - Prove Theorem 8.38: The change of variable x=ya4...Ch. 8.5 - Show that the change of variable in
yields an...Ch. 8.5 - Derive the quadratic formula by using the change...Ch. 8.5 - Use the definition of the discriminant
to show...Ch. 8.6 - True or False
Label each of the following...Ch. 8.6 - Label each of the following statements as either...Ch. 8.6 - Label each of the following statements as either...Ch. 8.6 - Each of the following polynomials is irreducible...Ch. 8.6 - In each of the following parts, a polynomial p(x)...Ch. 8.6 - In Exercises 36, a field F, a polynomial p(x) over...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 - In Exercises , a field , a polynomial over , and...Ch. 8.6 - For the given irreducible polynomial p(x) over 3,...Ch. 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 - Find the multiplicative inverse of in , where is...Ch. 8.6 - Find the multiplicative inverse of 9333+2 in (33),...Ch. 8.6 - An element of a field is a perfect square in if...Ch. 8.6 - Determine whether each of the following...Ch. 8.6 - Find the value of c that will cause the polynomial...Ch. 8.6 - Each of the polynomials in Exercises is...Ch. 8.6 - Each of the polynomials in Exercises is...Ch. 8.6 - Each of the polynomials p(x) in Exercises 1518 is...Ch. 8.6 - Each of the polynomials in Exercises is...

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