Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x) divides g(x) in Z[x].

Q: The number of irreducible monic polynomials of degree 2 over Z5

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Q: prove that if (xn) → 2, then (a) (2xn−1/3) → 1; (b) (1/xn) → 1/2. (For this exercise the Algebraic…

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Q: (a) Show that f(r) = x² + x + 4 is irreducible over Z11. (b) Write f(x) = x³ + x² + x + 1 as a…

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Q: For f(x), g(x), and Zn[x]in the given question ,, find the greatest common divisor d(x) of f(x) and…

A: Given: f(x)=x3+2x2+2 and g(x)=2x5+2x4+x2+2 in Z3[x]. To determine: the greatest common divisor d(x)…

Q: b. In Z6(x], let f = 2 + x + 4x² + 3x³ and g = 4 + 3x+ 5x² + 3x³. Compute f + g and f.g and their…

A: In the finite field we only use the coefficients from that field . The degree of polynomial is not…

Q: For an irreducible polynomial p(X) = 1+X + X² € F[X] consider F4 generated by p(X). Find x1, x2 E F,…

A: Sol

Q: Prove that the effective factor is a Hermitian factor in a specif and x-i

A: Momentum operator is defined as an Hermitian generation of translations, then it is hermitian by…

Q: Letf(x)= (x - 1)(x - 2)(x - 3) Prove that there is more than one 'c' in (1, 3) such that f' (c) =0.…

A: We have to find the value of c

Q: Show that for all x, e* > 1 +x.

A: It is given that, ex≥1+xwe have to show that for all x.

Q: #: Let X₁, X₂, ..., X₁ ~ f(x) = {0x0-1 0<x< 1 find Suff. Comp. St. " e.w.

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

Q: Prove that the function f:Q Q given byf(x) = 2x – 3 for all x E Q is bijection

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Q: -3n2 + 2n 2n2 + 4n +7 Compute the limit of (an) and prove it where an %3D

A: Evaluate an1∞ Where an=-3n2+2n2n2+4n+7

Q: a+ib a) If x+ iy = prove that x? + y² = 1. a-ib

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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 T(x1,x2)=(2x1-2x2,9x1-8x2). SHow that T is invertible, and find a formula for T^-1(x

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Q: Find infinitely many polynomials f(x) in Z3[x] such that f(a) = 0 for all a in Z3.

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Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

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Q: (b) Let g(x)=√x with domain {x|x ≥ 0}, and let > 0 be given. For each c> 0, show that there exists a…

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Q: The number of reducible monic polynomials of degree 2 over Zz is

A: The number of reducible monic polynomials of degree 2 over Z3 is

Q: Let g(x) =−x2 +4x+1−x2 +4x+1 Find when:   g(2), and  g(t)

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Q: Suppose f (x) =x²csc²(x³). Evaluate: A - 3cot(x³) + C B -con(x*) +C © -x²cot(x³) + C O -co(x*) + C

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Q: Suppose that f(x)=A/x,     g(x)=Bx^3, and    h(x)=ax^2+bx+c   Assume A≠0. Compute the composition…

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Q: Let f(x) = 2x +3, g(x) = x – 4, and h(x)= 4x – 1. Compute (h o go f)(3).

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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: If (an) has a decreasing subsequence, then (an) does not tend to infinity.

A: False

Q: Let f and g be the functions.

A: The composition of fogx is defined for all value of x for which x is in the domain of

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: Let A be invertible. Prove that At is invertible and (At)−1 = (A−1)t.

A: Given:Let A be invertible. Prove that AT is invertible and (AT)−1 = (A−1) T.

Q: Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions. |x² w(x², x5)= 2x

A: Wronskian is a deteminant which is test for proving whether solutions are linearly independent or…

Q: Let Qc(x) = x2 + c. Prove that if c 1 such that Qc is topologically conjugate to Fµ(x) = µx(1 − x)…

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Q: Prove that is G1 is isomorphic to G2 and H1 is isomorphic to H2, then G1 x H1 is isomorphic to G2 x…

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Q: Which of the following is indeterminate at x = 1? x² + 1 x2 – 1 x2 – 1 Vx +3 - 2 x2 +1 Vx + 3 – 2 x…

A: We plug x=1 in each expression

Q: Represent algebraically and graphically the set of values of z for which |z+3| = |2 – 2i|.

A: Solution

Q: Let ƒ(x) = x - 3, g(x) = √x, h(x) = x3, and j(x) = 2x. Expresseach of the functions as a composition…

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Q: Prove that for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

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Q: h(x)=-5-5x.

A: Given polynomial is                               h(x) = -5 - 5x

Q: In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer). 1) Prove that I is an ideal of Z[x]. 2)…

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Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

A: No it doesn't follow the statement

Q: Let x ∈ R. Prove that if x3 − x > 0 then x > −1.

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Q: Prove that f(x) = x2 + 2x is not injective.

A: The function is said to be injective if f(x)=f(y) it implies x=y

Q: 2.3 Theorem. Let p(x) be an irreducible polynomial in F[x). Then there exists an extension E of F in…

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Q: Letf(x) = (x - 1)(x - 2)(x - 3) Prove that there is more than one 'c' in (1, 3) such that f' (c) =…

A: We have to find the value of c

Q: Let A = k 0 1 k - 1 2k k+ 1. Use cofactor expansion to determine which values of k make A…

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Q: Compute x* + х'+x*+x'+x+1* (х'— х* +х' - х) over GF(2)

A: We have here 2=0 and 1=-1 due to field property.

Q: z, Z2 Prove

A: If z =x+ iy then  z = x- iy. i2 =-1.

Q: A function is invertible if and only if it is a bijection. True or False?

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Q: Find the linearization L(x) of f(x) at x = a. f(x) = 3square root of x, a = 27 show all work

A: fx=3x, a=27 Differentiating with respect to x, we get f/x=3·12x    ∵dxdx=12x=32x Now,…

Q: Given a closed set A, construct a continuously differentiable func- tion that has A as its set of…

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Q: Let ƒ(x) = x - 3, g(x) = √x, h(x) = x3, and j(x) = 2x. Expresseach of the functions as a composition…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.