F be a field and F be a subfield E. The mapping : F[X]-> E defined by af₁ = f(a) called the evaluation homomorphism. ven the evaluation homomorphism $:C[x]-> C₁ aluate the given polynomial below. ote that (i)(i) =i² = -1. 2 Pi (2x²³²-x² + 3x − 2) where i = -1

Q: Q2: (a) Determine which of these functions are injective, surjective, and bijective. The function f…

A: a:Given are the following functions:i fx=2x+1ii fx=exiii fx=x3-3xTo find that the following…

Q: Consider Z[x] = {a,x" + a₁₁x¹+...a₁x+ao: a; € Z,i = 0,1,2,...,n), the set of all polynomials in…

A: Inverse elements are 1 and -1

Q: 1. Find the first, second, third and fourth order Maclaurin polynomials of f(x) = arctan(x). Draw…

A: f(x)=tan-1x To find the first, second, third and the fourth order Maclaurin polynomial of f(x) To…

Q: f(x) = ao + a,x + azx² + azx³ € R[x] polinomu için –2 e R, f(x) polinomunun bir kökü ise x+2 e R[x]…

A: As ℝ is a ring of real numbers. So ℝ[x] is the polynomial ring that contains all polynomials in x…

Q: Consider the function f(x) = x + 2/1. Construct the Lagrange interpolation polynomial which…

A: The given function is f(x)=x+2x. The node points are x0=12, x1=1, x2=2. Calculate the value of the…

Q: We denote the set of polynomials with real coefficients as R[X] := {ao + a₁X + a₂X² + ... + a₂X": n…

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Q: If f(x) is a polynomial of degree <3, f(a) a x-a

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Q: Find a monic polynomial of least degree in R[x] that has 1-i and 2i as roots. write the powers of x…

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Q: Let a E K and let f(x) E K[x]. Prove that the remainder when f(x) is divided by x – a is ƒ(a).…

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Q: Let P(x) be the Lagrange interpolating polynomial that fits the data(-1,2); (4,2) and (7,5). Then…

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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: Given a function f (x). Suppose that, the Newton's interpolating polynomial P2(x) of f (x) at the…

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Q: Let P(x) be the Lagrange interpolating polynomial of 3+3 at the points 0 I and a 3 If P(x) ed to spi…

A: We will first find the Lagrange polynomial

Q: Let f and g be the functions defined by f : R → R+ ∪ {0} with f (x) = x2 and g : R+ ∪ {0} → R with…

A: (fog)(x) is the value of the function f(x) at g(x) (fog)(x)= f(g(x))

Q: 2. Let R be a U.F.D. and f (x), g (x) e R [x]. If f(x) g (x) is a primitive polynomial, then f (x)…

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Q: Let f: Z/4Z → Z/2Zx Z/2Z be defined by f([0])= ([0], [0]), ƒ([1]) = ([1],[1]), f([2]) = ([0], [1]),…

A: We see,         [2] = [1] + [1]    but, f([2]) = ([0], [1])          f([1]) + f([1]) = ([1], [1]) +…

Q: Represents the ff.on a number line and give its value. Examples: /-7/=7 /4/ = 4 -10 -9 -8 -7 6 5 4…

A: Topic - number line

Q: Obtain the Lagrange interpolating polynomial for a certain function f of which we know that: f (-1)…

A: Lagrange interpolating polynomial: The Lagrange interpolating polynomial is a degree polynomial that…

Q: 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)…

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

Q: Derive the autocorrelation function of: x₁ = $xt-1 + εt + 0€t-1 assuming that p < 1, and E(?) = 0²,…

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Q: Find Ao, A1, and Az in rewriting the function of f(x=2vx 0マx in terms of Lagure polynomial ョーの

A: Given: Let us consider the given function  fx=2x       x≥00             x&lt;0 and the Lagure…

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

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Q: Z Prove that the residue of functions and 2³ - 7² +1 Z3 (z-a) (z-b) at infinity are -1 and 1…

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Q: 2. Let char F = 0 and f(x) E F[x] be a monic polynomial having positive degree. If d(x) = gcd(f(x),…

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Q: 3. Let P(x) be a polynomial of degree n. If n is even, prove P(x) cannot have an inverse function.

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Q: Let G be a direct product of H and k product of 2 (H) 'and z (k) then Z(G) is a direct

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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: Let P(x) be the Lagrange interpolating polynomial that fits the data (-2,3); (2,4) and (4,3). Then…

A:

Q: Let ro, r1,..., r10 be 11 distinct values. Let L10,i(x) (i = %3D 0,..., 10) be the polynomials for…

A: Lagrangian polynomial: If x0,x1,...,xn are any n+1 distinct numbers, then for i=0,1,...,n, the i-th…

Q: (a) Let f, 9, h : N → R²º. Prove or disprove that if f + g€ O(h), then f e O(h) and gE O(h). (b) For…

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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…