Numerical2 for1. Let R(m) be the number of zeros of x² – 1 in Zm. Then R(p)every prime p > 2. Compute R(m) for m < 10. Investigate R(p¹) forprime p. Is R a multiplicative function? Does this work for otherpolynomials in place of a2 - 1?=

To compute R(m) for m < 10, let's find the number of zeros of x^2 - 1 in Z_m for each m:1. For m…

Answered: 1. Let R(m) be the number of zeros of ²…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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50 minul cteristics in 27. A fourth degree polynomial, P(x), with real coefficients has 4 distinct zeroes. Two of these zeros are -5 and 21 Which of the following is a logical conclusion about the other zero(s)? A. The other zeroes must be 5 and-2i. B. One of the other zeroes must be -5 ± 2i. C. One of the other zeroes must be 5 D. The other zeroes must be -2i and a real number. 28. The number of hours of daylight, y, in Utica is a function of the number of days, x, since January 1st 2013: y = 3.06 sin(0.017x – 1.40) + 12.23 How many hours of daylight, to the nearest tenth, does the model predict for February 14th, 2013? A. 9.4 В. 10.4 C. 11.7 D. 12.2
1. Compute the resultant of x polynomials have a common factor in Q[x]? Explain your reasoning. - 3x* – 2x + 3x² + 7x + 6 and x +x² + 1. Do these
3
Qu6. Consider the polynomial below in which we have a parameter z. Q(x)=x2³+(2-5)x² + (z+1)x+ (11-32). Find the set U of all real numbers for which the polynomial Q(r) satisfies the Routh- Hurwitz criteria.
(1) For each member of your group record their response to: “ Give an example of a polynomial and an expression involving x that is not a polynomial." (2) A box with no top is to be fashioned from an 18 inch x 24 inch piece of cardboard by cutting out congruent squares from each corner of the cardboard and then folding the resulting tabs. Let x denote the length of the side of the square which is removed from each corner. Determine a polynomial which gives the vol- ume of the box as a function of x. Use a plot of this polynomial to determine the optimal value of x.
4. For this problem, you may take for granted that 47 and 53 are prime, and that 2491 = 47.53. Consider the polynomial f = X²+3. Justify your answer for each part. (a) Determine whether or not the polynomial X² + 3 has a root in F47. (b) Determine whether or not the polynomial X2 +3 has a root in F53- (c) Determine whether or not the polynomial X² + 3 has a root in Z/2491Z.
Which of the following polynomials is reducible over Q: x3 +2 x4 + 3 x5 + 1 x2 + 3 I thought it was x4 + 3 but that was wrong. Is it x5 + 1?
8. Let F = Z3. Factor the polynomials x³ + x² + 2x + 2 and x³ + 1 into products of primes in F[x], and give their greatest common divisor.
-5 -4 -2 7,90 711 632 553 474- 395 316 237 158 79 23
(3.2) #4i
Jj6.
Given the following set S of polynomials , find a polynomial p(x) in P2 so that SU {p(x)} spans P2 and a non-zero polynomial q(x) in P2 so that SU{q(x)} does not span P2. Use the 'A' character to indicate an exponent, e.g. 5x^2-2x+1. S = -8x2+9x-9, 5x²+5x %3D p(x) = 0 q(x) = 0 %3D

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1. Let R(m) be the number of zeros of ² - 1 in Zm. Then R(p) = 2 for every prime p > 2. Compute R(m) for m < 10. Investigate R(p) for prime p. Is R a multiplicative function? Does this work for other polynomials in place of x² - 1 ?