The ring Z[/2] is an ED with d(a + bv2) = |a² – 21³| . Show that the equations r² – 2y² = 1 and r² – 2y² = -1 cach have infinitely many integer solutions.

Q: Find the inverse of X + (X² – 8X + 7) in the form aX + b in the ring Q[X]/(X² – 8X + 7). - a b =…

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Q: Let A=[124] Recall that row(A) and nul(A) are orthogonal complements. So we can split any x E R²…

A: Since the question has multiple subparts, as per the company guidelines, we will solve the first…

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: Find Tay br Polynomial. for the function fox): Vx + 3. ot degree 3 centered ata=2

A: According to the problem, we have

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: 4. Consider the complex number w = 3cis (). For which value(s) of k is z - w a factor of the…

A: We will use remainder theorem to find the value of k. Remainder theorem says that if x-a is a factor…

Q: {6 ):a € Z,6b,c e Q}. Together with matrix addition and multiplica- Let R= tion, show that Ris a…

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Q: 2. Find all of the prime quadratic polynomials in F4[x], where F4 {0,1, a, b} be the field of four…

A: Given: F4={0,1,a,b}

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: If A²=0, prove that A cannot be invertible.

A: Given,If A²=0, prove that A cannot be invertible.

Q: 2. Assign Hermit's interpokting tolynomial by The fallowingy da ta:

A: The Hermite's interpolation polynomial is H(x)=∑ui(x)⋅yi+∑vi(x)⋅yi'. Here…

Q: . Let 21 2i and z2 = 1+v3i. (a) Express zi and z2 in mod-arg form. (b) Plot in the complex plane the…

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Q: 1. Let B₁ = = {-1} {[]} i) Find Rep ii) Find Rep B1 and -{[]} B2 = be two ordered bases for R² C…

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Q: Find the conjugate of z = -2 – 4i, and compute zz. z = zz =

A: Given that: z=-2-4i

Q: If 7,=-4-J5 Z2= -3 +j2, and Zg =2-3, what is Re (23-2 7, + 23 Note : z* is the complex conjugate Of…

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Q: Find all solutions of x2 + x = 0 in the ring Z12

A: Given x2+x=0

Q: Solve the following using Cramer's Rule, 5x + y – z = -7 2х — у — 2z %3D 6 3x + 2z = -7

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Q: Show that is isomorphic to Z2.

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Q: Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is…

A: Given : The operation on ℝ2 is defined as  x, y = x1y1 - x1y2 - x2y1 + 3x2y2 ,  where x = (x1 , x2)…

Q: 3. How many units are in the ring (Z/15Z) × (Z/125Z)?

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Q: 2) The complex conjugate of z is denoted as z. Solve for z in the equation: Z – 12 = j(9 – 27) -…

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Q: Let T:P2(R) → P2(R) T(a + bx + cz?) = 3a – 16 + 6c + ( – 1la + 4b – 22c)z + (– 6a + 36 – 1lc)z? Let…

A: Let us us consider the mapping T:P2ℝ→P2ℝ be defined by Ta+bx+cx2=3a-b+6c+-11a+4b-22cx+-6a+3b-11cx2…

Q: Let z be a root of z5 – 1 = 0 with z # 1. Compute %3D 215 + 216 + z17 +· . + z50

A: Since z is a root of z5-1=0 ⇒z-11+z+z2+z3+z4=0⇒1+z+z2+z3+z4=0…

Q: For an integer t, prove that gcd(2t + 1, 9t + 4) = 1.

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Q: 2. Complex Numbers: Show that for any two complex numbers, A₁ = a₁ + b₁j and A₂ = a₂ + b₂j, where…

A: The objective of this question is to prove the properties of the modulus and argument of the product…

Q: Show that over the ring Z/30Z[x] the polynomial f(x)=x has the factorization…

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Q: Show that <a, b | a5 = b2 = e, ba = a2b> is isomorphic to Z2.

A: We have to show that <a, b | a5 = b2 = e, ba = a2b> is isomorphic to Z2.

Q: Consider the set C of complex numbers. Let (c², d) be a function where c² represents an element from…

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Q: Let α be a zero of x3 + x2 + 1 in some extension field of Z2. Solve the equation (α + 1) x + α = α2…

A: Given that α be a zero of x3+x2+1 in some extension field of ℤ2. We have to solve the equation…

Q: i. Show that 3 + 4i and 4-3i are associates in Z[i]. ii. Determine the splitting field of x4 + x2 +…

A: (i).

Q: Use the standard basis for P2, B = {1, t, t²}, and the coordinate mapping to show that 10 – 12t –…

A: The solution is below.