Suppose that p and q are integers with 0 < p, 0

Q: a) Find a monic polynomial in (Z/8Z) [x] with more distinct roots (in Z/8Z) than its degree. b) Find…

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Q: (d) (x² + 2x + 1)q(x) = 1 (mod x³ + x² + 1) over Z3

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Q: QUESTION 2 Show how to find the cyclotomic polynomials Þ(x) for 1 ≤ n ≤ 15.

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Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

Q: IF TE A (V) and if 9 (1) € Fcx) is such that 9 (T) = 0, a characteristic root it true that every…

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Q: IF TE A (V) and if 9 (1) € Fox) is such that a (T) = 0, is a characteristic root it true that every…

A: The statement is false.

Q: For the data O True 0 0.3 0.6 0.9 1.2 f (x) 1 1.8221 3.3201 6.0496 11.0232 where the function f is…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: (b) Consider the following set of real numbers: 3n² +n +3 A-3+N} CR { n³ n = :n Prove that A is…

A: (b) Given that (i) Noted that So we have 0 is lower bound and is upper bound of A .Hence A is…

Q: Let Br(X) denotes k-th Bernoulli polynomial and B be the k-th Bernoulli

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Q: Suppose that the function f(x)  is continuous everywhere. Suppose further that the points (Xj, Yj),…

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Q: Show that for every positive integer n there exists an irreducible polynomial of degree n over Fp.

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Q: Obtain a formula for the polynomial p(x₁) = yi p of least degree that takes these values: p'(x₁) = 0…

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Q: ;0<x<0, - 0 <0 <0 Let X,,X,X,r(x;e)=- e.w Discuss the sufficiency by using Factorization Theorem.

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Q: 7. Let a be a nonzero real number. Evaluate lim o f (x) and lim,→ f(x), where -00 f (x) = a.x° – 3x*…

A: Given problem is :

Q: as x 1 Let #E(³ (R), such that ‡ (x) = 2(x-1) + 3 (x-1)³ + 0((x-1)³) It's true that A. (2) = 5 B. ¹¹…

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Q: Let på(x) be a polynomial that interpolates the function f(x) = ½1⁄2 at n + 1 distinct nodes on the…

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Q: 2.6 Let f(x) = x² + ax+bZ[x] be a quadratic polynomial with integer coefficients, for example, f(x)…

A: This is an open problem and is quite hard to solve. The details are given in step 2.

Q: 4. Find a polynomial p(x) of degree at most 2, which is not in the span of P₁(x) = x² - 2x +1,…

A: p1x=x2-2x+1p2x=x-1p3x=x2-1

Q: 3. Suppose f(x) and g(x) are monic integer polynomials. Prove that f(x)|g(x) in Q[x] if and only if…

A: Integer polynomials are polynomials having integer coefficients. And the monic polynomials have…

Q: Suppose that f (x) and g(x) are irreducible over F and that deg f (x)and deg g(x) are relatively…

A: We have to show that g(x) is irreducible over F(a).

Q: Let Q(x) be any polynomial of degree ≤ n with real coefficients and let M be the maximum of Q(x)| on…

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Q: Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such…

A: Given K be any field and a1,a2,...,an be pairwise distinct elements of K. For each i=1,2,3,...,n, pi…

Q: 1, Give a complete proof of the growth rate of the polynomial below. You should provide specific…

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Q: Prove that for any polynomial P of degree <n, | P' (x) | < n(1 || P || - x²)-1/2 || P ||, where max…

A: We will construct another function in terms of P(x) and prove the given statement.

Q: Use the Fibonacci algorithm to find the minimum of 1 2(x- 1) log r- (x – 1)2 f(r) = I+1 which is…

A: From the given information. The function for which the minimum has to be calculated is as follows.

Q: Let n P₁ P2 = ppp be the prime factorisation of an integer n > 1. 1: ti A 11. C.

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Q: Let g(x) be a polynomial in Z₂[x]. Prove that if the polynomial code C generated by g(x) with length…

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Q: Using Neville's algorithm, calculate P2345 for T = 3.0 K. Refer to ta table above.

A: As per the question we are given a data table abou the temperature (K) and the vapour pressure (kPa)…

Q: For each positive integer n, there exists an irreducible  polynomial f(x) of degree n over…

A: The statement you mentioned is a result known as the Existence of Irreducible Polynomials over a…

Q: For each positive integer n, there exists an irreducible polynomial f(x) of degree n over…

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Q: For any n> 1, let (n) be the number of positive integers a <n with gcd(a, n) = 1. This function is…

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Q: Find the absolute maximum and the absolute minimum of U(q) 20002 (15 5q)4 on the nterval 04.

A: First find the derivative of the given function and compute the critical points.