Let Rz] be the set of all polynomials in variable x Let R(x) be the set of all rational functions over R, that is, R(1) := {p(¹) Note that R[x] R(x) and that R(x) is a field. : p(x), q(x) = R[x], q(x) ‡ 0 (x) #0}. Let A (r) be a 3 x 3 matrix with entires from R(x) given by G -X Dotorming if 4(r) is nonsingular/invertible over R(x), and if so, then find A-¹(x). A(x) = X 2+x -1+x² -3+3x+x² x² -1-x-x³ -2x - x² - 7³

Q: Show that: f(x) = x² + x + 2 € Z₂[x] is irreducible. Further, find the order of f(x).

A: Irreducible polynomial and it's order

Q: Show that f(x)=x\power{2}-3 is the minimal polynomial of \sqrt{3} over Q,

A: Let            We  have to  find the  minimal  polynomial of  the    over Let Then,  Suppose  that

Q: Let f(x)= x³ + 2x+ 1 and g(x)= 4x + 1 be two polynomials in a ring (Zs[x],+ ,.). Find q(x) and r(x).

A:

Q: The function

A: Given, the function f : ℂ→ℂ defined by                   f(z)=ez+e-zhas

Q: Let R[x] be the set of all polynomials with real coefficients. In other words, x] = {anx" + an-1x"-1…

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Q: The coordinates of the P(x)=x³ – 2x2 + 3x polynomial with respect to the base B-[1, x, x?,x'] are as…

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Q: Let f(z) be an entire function satisfying |ƒ (²1 +22)| ≤ |ƒ (2₁)| + + + |ƒ (2₂) | = n for all 2₁, 22…

A: As per the question we are given an entire function which satisfies the following inequality :And…

Q: Find an equation of the plane passing through the given points. (7, 8, –8), (7, -8, 8), (–7, –8, –8)

A: Given problem is :

Q: Let f: Z → Z be defined by f(x)=x²-1, and let C = {0, 1, 3). Find f(C), f-¹(C), f(f(C)) and…

A: Here we will find the given function on finding the inverse of the given function and after that we…

Q: , Consider the interpolating polynomial for f(x) =x² based on th points (0,0). (1,1) and (2,8). Find…

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Q: Let A = R {3}, B = R - {1}, and let 1 x-2 Isfinvertible? Explain. x-3 f: A B be defined by f(x)=

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: Show that there are no polynomials f(x) and g(x) in Z[x] such that 1 = 2f(x) + xg(x)

A:

Q: Let & be a zero of f(x)=x2 +2x+2 in some extension field of Z3. Find the other zero of f (x) in…

A: Consider the given function fx=x2+2x+2. Note that, as the given function is quadratic, the function…

Q: Suppose f(z) is an analytic function with domain C (the set of all complex numbers). Suppose further…

A: Suppose further that in the region {|z| > 100} f satisfies f(Z)<a+ b |z|^n where a ,b are…

Q: Prove that F (3.x)(A → B) → (Vx)A → (3x)B. 3.

A: To prove: ⊢∃xA→B→∀xA→∃xB

Q: Let x0, x1, ..., Xn be distinct points on the interval [a, b], where Xo <x₂ < x₂ <--- < x₂ and let f…

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Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

A:

Q: Let f(x)= x3 + x2 + x +1  ∈ Z2[x]. Write f(x) as a product of irreduciblepolynomials over Z2.

A: Given polynomial is: f(x)=x3+x2+x+1 Since, f(1)=0 So, x-1 is a factor. Now, factorize the…

Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. b) Prove that f is differentiable…

A: Given fx=x2x is rational0x is irrational To prove f is differentiable at exactly one point, namely…

Q: Prove that (Z(sqrt(2)), +, ×) is an integral domain.

A: We know that , ℤ2 = a + b2 / a , b∈ℤ . Clearly , ℤ2 forms a ring. We know that , a commutative ring…

Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

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Q: Let f be some continuous function on an open interval I containing x0, x1 and x2, where xo <x1 <x2.…

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Q: 3. Let x0, x₁,...,xn be distinct points on the interval [a, b], where Xo <x₁ < x₂ <... < xn and let…

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Q: an

A:

Q: Suppose P(z)= ao+a₁z + a₂z²+ ... + anzn is a polynomial of degree n. Show that for each 0 k≤n, the…

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Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

A:

Q: Let zo + 1 be a zero of order 3 of an analytic function g(z) and f (z) = (z-1)2 (g(z))³" Then z, is…

A: Use the concept that if f(z) has zero at z0 of order m then 1/(f(z)) has pole at z0 of order m. By…

Q: Prove where the following complex function is analytic

A:

Q: f(x) = x² − 3 is the minimal polynomial of √3 over Q,

A: We have that              f(x) = x2-3  be a polynomial of two degree.

Q: ƒ(X) e F[X] be a non-constant polynomial of degree d. Prove that f(a) has degree at least over F.

A: ANSWER :