19. We say that a e Fis a multiple root of f(3) = FE (- a* isa factor of x) for some kz 2. ) Prove that a eR is a multiple rootof ftx) E Rl if and only if e is a root of both f(x) andƒ(x), where f"(x) is the derivative of f(x). (6) T (*) E RE and if E) is relatively prime to S (8), prove that /(x) has no multiple root in R.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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#19 from textbook

b My Questions | bartleby
Thomas W. Hungerford - Abstrac ×
+
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
A Read aloud V Draw
F Highlight
O Erase
133
of 621
(b) Supposc r, s, t e Farc roots of ax + bx + ex + de F[x] (with a + Op).
Show that r + s + t = -a-b and rs + st + rt = a-'cand rst = -a'd.
15. Prove that +1 is reducible in Z[x] if and only if there exist integers a and b
such that p = a + b and ab = 1 (mod p).
16. Let f(x), g(x) E F[x] have degree sn and let co, C1, . . . , C, be distinct elements
of F. If f(c) = g(c) for i = 0, 1, ..., n, prove that f(x) = g(x) in F[x].
17. Find a polynomial of degree 2 in Zdx] that has four roots in Z. Does this
contradict Corollary 4.17?
18. Let ç:C → C be an isomorphism of rings such that o(a) = a for each
a E Q. Suppose r e C isa root of f(x) E Q[x]. Prove that o(r) is also a
root of f(x).
19. We say that a E Fis a multiple root of f(x) E F[x] if (x - a)* is a factor of
f(x) for some k > 2.
(a) Prove that a ER is a multiple root of f(x) e R[x] if and only if « is a
root of both f(x) and f'(x), where f"(x) is the derivative of f(x).
(b) If f(x) E R[x] and if f(x) is relatively prime to f"(x), prove that f(x) has
no multiple root in R.
20. Let R be an integral domain. Then the Division Algorithm holds in R[x]
whenever the divisor is monic, by Exercise 14 in Section 4.1. Use this fact to
show that the Remainder and Factor Theorems hold in R[x].
21. If R is an integral domain and f(x) is a nonzero polynomial of degree n in
R[x], prove that f(x) has at most n roots in R. [Hint: Exercise 20.]
22. Show that Corollary 4.20 holds if Fis an infinite integral domain. [Hint: See
Exercise 21.]
23. Let f(x), g(x), h(x) E F[x] and re F.
(a) If f(x) = g(x) + h(x) in F[x], show that f(r) = g(r) + h(r) in F.
(b) If f(x) = g(x)h(x) in F[x], show that f(r) = g(r)h(r) in F.
Where were these facts used in this section?
24. Let a be a fixed element of Fand define a map 4.¿F[x] →Fby Pa[f(x)] = f(a).
Prove that o, is a surjective homomorphism of rings The map q, is called an
evaluation homomorphism; there is one for each a e F.
25. Let Q[T] be the set of all real numbers of the form
ro + riT + r2 + ·.·+ a,", with n20 and r, E Q.
(a) Show that Q[T]is a subring of R.
(b) Show that the function 0:Q[x] –→Q[7] defined by ®( f(x)) = f(T) is an
isomorphism. You may assume the following nontrivial fact: 7 is not
the root of any nonzero polynomial with rational coefficients. Therefore,
Theorem 4.1 is true with R = Q and T in place of x. However, see
Exercise 26.
2:59 PM
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EPIC
11/20/2020
Transcribed Image Text:b My Questions | bartleby Thomas W. Hungerford - Abstrac × + O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf A Read aloud V Draw F Highlight O Erase 133 of 621 (b) Supposc r, s, t e Farc roots of ax + bx + ex + de F[x] (with a + Op). Show that r + s + t = -a-b and rs + st + rt = a-'cand rst = -a'd. 15. Prove that +1 is reducible in Z[x] if and only if there exist integers a and b such that p = a + b and ab = 1 (mod p). 16. Let f(x), g(x) E F[x] have degree sn and let co, C1, . . . , C, be distinct elements of F. If f(c) = g(c) for i = 0, 1, ..., n, prove that f(x) = g(x) in F[x]. 17. Find a polynomial of degree 2 in Zdx] that has four roots in Z. Does this contradict Corollary 4.17? 18. Let ç:C → C be an isomorphism of rings such that o(a) = a for each a E Q. Suppose r e C isa root of f(x) E Q[x]. Prove that o(r) is also a root of f(x). 19. We say that a E Fis a multiple root of f(x) E F[x] if (x - a)* is a factor of f(x) for some k > 2. (a) Prove that a ER is a multiple root of f(x) e R[x] if and only if « is a root of both f(x) and f'(x), where f"(x) is the derivative of f(x). (b) If f(x) E R[x] and if f(x) is relatively prime to f"(x), prove that f(x) has no multiple root in R. 20. Let R be an integral domain. Then the Division Algorithm holds in R[x] whenever the divisor is monic, by Exercise 14 in Section 4.1. Use this fact to show that the Remainder and Factor Theorems hold in R[x]. 21. If R is an integral domain and f(x) is a nonzero polynomial of degree n in R[x], prove that f(x) has at most n roots in R. [Hint: Exercise 20.] 22. Show that Corollary 4.20 holds if Fis an infinite integral domain. [Hint: See Exercise 21.] 23. Let f(x), g(x), h(x) E F[x] and re F. (a) If f(x) = g(x) + h(x) in F[x], show that f(r) = g(r) + h(r) in F. (b) If f(x) = g(x)h(x) in F[x], show that f(r) = g(r)h(r) in F. Where were these facts used in this section? 24. Let a be a fixed element of Fand define a map 4.¿F[x] →Fby Pa[f(x)] = f(a). Prove that o, is a surjective homomorphism of rings The map q, is called an evaluation homomorphism; there is one for each a e F. 25. Let Q[T] be the set of all real numbers of the form ro + riT + r2 + ·.·+ a,", with n20 and r, E Q. (a) Show that Q[T]is a subring of R. (b) Show that the function 0:Q[x] –→Q[7] defined by ®( f(x)) = f(T) is an isomorphism. You may assume the following nontrivial fact: 7 is not the root of any nonzero polynomial with rational coefficients. Therefore, Theorem 4.1 is true with R = Q and T in place of x. However, see Exercise 26. 2:59 PM O Search for anything EPIC Ai EPIC 11/20/2020
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