(c) x + 4x + 2x + 3x -x+5 (4) x + Sx + 4x + 7 19. Write euch polynomial as a product of irreducible polynomiałs in Q[3).
(c) x + 4x + 2x + 3x -x+5 (4) x + Sx + 4x + 7 19. Write euch polynomial as a product of irreducible polynomiałs in Q[3).
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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Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Thomas W. Hungerford - Abstrac ×
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T2. Let F be a neid and jX) E FXI. II CE Fand JX + C) IS Irreaucibie in F|X],
prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.]
13. Prove that f(x) = xt + 4x + 1 is irreducible in Q[x] by using Eisenstein's
Criterion to show that f(x + 1) is irreducible and applying Exercise 12.
14. Prove that f(x) = x* + x + + x + l is irreducible in O[x). [Hint: Use the
hint for Exercise 21 with p = 5.]
15. Let f(x) = a,t + an-1x1 +• ·•+ a,x + a, be a polynomial with integer
coefficients. If p is a prime such that p|a1, p|az, ... ,P|a, but p } m, and
Ot 2012 C Le A Rig taved May at be opind cd ar we ar la pt Dete drie d perty cot y bepd me Bootendtr C o. Edalvew ba
dd t oy ppd t dan at ty het he ovlrgpert Cgge Laig ma right tomveddonl at y tme i dghs anmguire
120 Chapter 4 Arithmetic in F[x]
pta, prove that f(x) is irreducible in Q[x). [Hint: Let y = 1/x in f(x)/x"; the
resulting polynomial is irreducible, by Thcorem 4.24.]
16. Show by example that this statement is false: If f(x) E 7[x] and there is no
prime p satisfying the hypotheses of Theorem 4.24, then f(x) is reducible in Qx].
17. Show that there are n+1 - n polynomials of degree k in Z[x].
18. Which of these polynomials are irreducible in Q[x]:
(a) メ-+1
(c) x + 4x + 2x+ 3x - x + 5
(b) x* +x +1
(d) x + 5x + 4x + 7
19. Write each polynomial as a product of irreducible polynomials in Q[x].
(a) + 2x – 6x? – 16x – 8
() x" – 2x – 6x– 15x – 33x – 9
20. If f(x) = a, +..+ a,x + ao, 8(x) = b,x + ...+ b,x + bo, and h(x) =
cxt + ...+ cx + co are polynomials in Z[x] such that f(x) = g(x)h(x), show
that in Z[x], F(x) = (x). Also, see Exercise 19 in Section 4.1.
C.21. Prove that for p prime, f(x) = x-1+ -2+ . ..+ x² + x + 1 is irreducible
in Q[x]. [Hint: (x – 1)f(x) = x – 1, so that f(x) = (x – 1)/(x – 1) and
f(x + 1) = [(x + 1y –11/x. Expand (x + 1)® by the Binomial Theorem
CoursSa
(Appendix E) and note that p divides
when k > 0. Use Eisenstein's
Criterion to show that (x + 1) is irreducible; apply Exercise 12.]
EXCURSION: Geometric Constructions (Chapter 15) may be covered at
this point if desired.
4.6
Irreducibility in R[x] and C[x]*
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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
142
of 621
T2. Let F be a neid and jX) E FXI. II CE Fand JX + C) IS Irreaucibie in F|X],
prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.]
13. Prove that f(x) = xt + 4x + 1 is irreducible in Q[x] by using Eisenstein's
Criterion to show that f(x + 1) is irreducible and applying Exercise 12.
14. Prove that f(x) = x* + x + + x + l is irreducible in O[x). [Hint: Use the
hint for Exercise 21 with p = 5.]
15. Let f(x) = a,t + an-1x1 +• ·•+ a,x + a, be a polynomial with integer
coefficients. If p is a prime such that p|a1, p|az, ... ,P|a, but p } m, and
Ot 2012 C Le A Rig taved May at be opind cd ar we ar la pt Dete drie d perty cot y bepd me Bootendtr C o. Edalvew ba
dd t oy ppd t dan at ty het he ovlrgpert Cgge Laig ma right tomveddonl at y tme i dghs anmguire
120 Chapter 4 Arithmetic in F[x]
pta, prove that f(x) is irreducible in Q[x). [Hint: Let y = 1/x in f(x)/x"; the
resulting polynomial is irreducible, by Thcorem 4.24.]
16. Show by example that this statement is false: If f(x) E 7[x] and there is no
prime p satisfying the hypotheses of Theorem 4.24, then f(x) is reducible in Qx].
17. Show that there are n+1 - n polynomials of degree k in Z[x].
18. Which of these polynomials are irreducible in Q[x]:
(a) メ-+1
(c) x + 4x + 2x+ 3x - x + 5
(b) x* +x +1
(d) x + 5x + 4x + 7
19. Write each polynomial as a product of irreducible polynomials in Q[x].
(a) + 2x – 6x? – 16x – 8
() x" – 2x – 6x– 15x – 33x – 9
20. If f(x) = a, +..+ a,x + ao, 8(x) = b,x + ...+ b,x + bo, and h(x) =
cxt + ...+ cx + co are polynomials in Z[x] such that f(x) = g(x)h(x), show
that in Z[x], F(x) = (x). Also, see Exercise 19 in Section 4.1.
C.21. Prove that for p prime, f(x) = x-1+ -2+ . ..+ x² + x + 1 is irreducible
in Q[x]. [Hint: (x – 1)f(x) = x – 1, so that f(x) = (x – 1)/(x – 1) and
f(x + 1) = [(x + 1y –11/x. Expand (x + 1)® by the Binomial Theorem
CoursSa
(Appendix E) and note that p divides
when k > 0. Use Eisenstein's
Criterion to show that (x + 1) is irreducible; apply Exercise 12.]
EXCURSION: Geometric Constructions (Chapter 15) may be covered at
this point if desired.
4.6
Irreducibility in R[x] and C[x]*
3:03 PM
O Search for anything
口
EPIC
Ai
へ
EPIC
11/20/2020
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