let p be a prime and f(x) = (x-1)(x-²)..( efficient of this polymas them all the co-e distinct divisible by P.

Q: Prove that there is only one polynomial P3(x) among all polynomials of degree < 3 that satisfy the…

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Q: A positive integer N is said to be a congruent number if it is the area of a right triangle with…

A: Given : A Positive Integer Is Said To Be Congruent, If It Is The Area Of A Right Triangle With Sides…

Q: Factor x^3 − y^3 into a product of irreducible elements in Q[x, y] and prove that each of the…

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Q: Find an integer coefficient cubic that has x = 1 and x = 2+ /3 as zeros.

A: Given: The two zeros of a cubic polynomial are x=1 and x=2+3.

Q: Recall the property that (a, b) = 1 if and only if ax + by = 1 for some integer x, y. Use this…

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Q: Suppose X and Y have joint characteristic function (s, t). Show that, subject to the appropriate…

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Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

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Q: 9. (3 Points) 4 Given A= Al then it is: 6- None of the given choices Symmetric O Orthogonal…

A: The correct option is : None of the given choices.   The detailed solution is as follows below :

Q: Suppose s and t are distinct natural numbers. Show that lcm(s, t) + lcm(s+1, t+1) > 2st/(sqr(|s −…

A: Given: s and t are distinct natural numbersTo prove: lcm(s, t)+lcm(s+1, t+1)&gt;2sts-t

Q: Suppose that the prime factorizations of two positive integers x and y are expressed using a common…

A: The prime factorization is breaking down a number into the product of only prime numbers. These…

Q: 3. Let R-Z18. Determine the units and zero divisors of R giving the explicit unit or zero divisor…

A: We know that  no of units in ring Zn = phi(n), and  no of zero divisor of Zn = n - phi(n) -1

Q: Show that /(-1)-7(1) but there is Let f(x) 1 Show that /(-1) (1) but there is no number e in (-1, 1)…

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Q: (c) Let n be a positive integer and assume that there are integers a and b ab. Show that a < yn or b…

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Q: 22 (b): let A and B two ideals with unity of Commutative A+B= IR ving IR Such that Prove A.B = ANB.…

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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: 4. Find the degrees of the following extensions. (a) Q(√3) over Q. (b) Q(√3, i) over Q(i). (c)…

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Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

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Q: 4.a) Show that (1+2i) is a zero of the polynomial Qcx) = x²-6x*+3x -20. Hence , or otheruwise, write…

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Q: Which among the following is an elliptic PDE? 6Uxx + 7Uxy +9Uyy = 0 7Uxx+6Uxy + 5Uyy = 0 8Uxx +4Uxy…

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Q: 2 5= {(0, 0₂, 93) | DER₁, A₂ER₂, az ER₂ Z Show that sis subring of RxP₂Rz

A: We have to show that S is a subring of ring R1×R2×R3.

Q: f for ƒ : R - {2} → R- { 5} indicated by f (x) =(5x +1)/(x -2a), a bijection can be made from [0, 1)…

A: Bijective function: The domain is set A and the co-domain in set B relates elements of two sets A…

Q: вршио Show that a group by of ordag pq for district prime p is solvable.

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Q: Determine the numbers a such that x a (p) is solvable for p = 7, 11, and 13. %3D

A: Given, x3≡amod p At p=7 We know that suppose p is rational prime p≡1mod 3 then x3≡a mod p is…

Q: 5. Prove that p(mk) = mk-1p(m) for all positive integers m and k.

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Q: 20.* a. Let P(z) = 1 + 2z + 3z² +.. of P(2) are inside the unit disc. + nz"-1, By considering (1 –…

A: Proved by using Rouches theorem from complex analysis.

Q: Find all monic irreducible polynomials of degree 2 over Z

A: A polynomial of degree 2 in Z3 [x] is irreducible if and only if it has no roots in Z3.

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: Prove this statement: we have numbers {a1, a2, a3 .,a2n} which are even and positive, but 0 < {a1,…

A: As, we know that, A.M.≥G.M. ---(1) A.M. of numbers, a1,a2,a3...a2n=a1+a2+a3+a4...+a2nn G.M. of…

Q: For any positive integers m, n (

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Q: Show that the polynomial z5 + 14z + 2 has four zeros in the annulus </2 < 2.

A: Given

Q: Q.5 Find all entine functipns a with

A: Given: The given function is fz≤zez

Q: Consider the following metria A - Find the trunsformation to reuch a diagenal form

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Q: Show that t (T (n)) = 2 if and only if n = p9-1 for some primes p, q. %3D

A: T(n) is the divisor function which counts the number of divisors of a number.

Q: 1) i) Prove that if a and b are integers, with b>0, then there exist unique integers q and r…

A: A question in elementary number theory. We shall solve this using basic facts of divisibility.

Q: Prove that {4a + 126 : a, b E Z} = {4c: c € Z}.

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Q: List all cosets of (([1]3, [2]6)) in Z3 × Ze

A: