4) Let n-1 f(T) = T" +) a; · Ti i=0 be a monic polynomial of degree n > 1 with a; E Z for all 0 < i < n – 1. Show that if f(T) has a root in Q then it has also a root in Z.

Q: 3) Let f(T) e R[T] be a polynomial of degree > 3. Show that there exists polynomials g(T) and h(T)…

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Q: 4) Let F be a field. The derivation f' (T) € F[T] of a polynomial f(T) E a; · Ti e F[T] is defined…

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Q: If f(a) = 7 and f(b) = 0, then there exists a number c in the interval (a, b) such that f'(c) 7 a –…

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Q: Let p(x) be a polynomial that is divisible by (x-a)n where n∈ N and n >= 2. Prove that p'(x) is…

A: Here, given that the polynomial p(x) is divisible by (x-a)n. So, p(x)=(x-a)nq(x)...(1) Where, q(x)…

Q: Find the Hermite interpolating polynomial for the function f(x) = √x satisfying the conditions…

A: According to Hermite interpolation the error is given by:…

Q: Let fn, f be real functions on S2. Show ∞ ∞ ∞ {w: fn(w) + f(w)} = UNU {w: | fn(w) - f (w)| ≥ } }.…

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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: Show that the Chebyshev polynomials have thefollowing properties: Tm(Tn(x)) = Tmn(x)

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Q: Define F : R → R and G : R → Z by the following formulas: F(x) = (x^2)/3 and G(x) = ⌊x⌋ for every x…

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Q: 39.. Find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. 1 f(x) =…

A: We know that Taylor Expansion : f(x)=11+x=1-x+x2-x3+x4-.........∞.

Q: Given the functon /(x) Suppose that the Newton's interpolating polynomial P,(x) of f(x) at the…

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Q: Let the function f(x) = ex be approximated by an interpolating polynomial of degree n = 10 in the…

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Q: Form the interpolating polynomial of degree k that satisfies pG) (xo) = f(G) (xo) for 0<j< k.

A: To form: The interpolating polynomial of degree k that satisfies pjx0=fjx0 for 0≤j≤k

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: For the data O True 0 0.3 0.6 0.9 1.2 f (x) 1 1.8221 3.3201 6.0496 11.0232 where the function f is…

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

Q: 1. What degree are the following polynomials (3.a and 3.b) and monomials (3.c and 3.d)? (a) f(æ) =…

A: Note: According to bartleby we have to answer only first three subparts please upload the question…

Q: 11. a. For the function f(x) = (3x+2), let x, =-1, x, =0, x, =1. Construct interpolating polynomial…

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Q: Suppose f: R → R is defined by the property that f(x) = x + x² + x³ for every real number x, and g:…

A: We are given that, for all real number x,          f(x) = x + x2 + x3  And        (g o f)(x) = x…

Q: Show that the function f : [0, 1] →[0, 1] given by f(x) = (x −1)2 has a fixed point. Can you apply…

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Q: 1. Find the Newton-Hermite interpolation polynomial for f(x) with f(1) = 8, f'(1) = 16, ƒ(3) = 16,…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Let a(t), 2 < t< 8 be a positive function. Also, let the function 8(t) be defined as a'(t)/a(t) on…

A: We will find out the required expression.

Q: Let φ1(x) = 1, φ2(x) = x/|x|, φ3(x) = x^2 − 1/3 for −1 ≤x ≤1. Show that (φ1,φ2,φ3) form an…

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Q: 2. Let (r) be the function defined by V(x) = 1. Let 1+x. In(x) Vovo...(x) n-copies Note that…

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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 p be an arbitrary polynomial p(x) = a„x" + a,-1x"-1 +..+a,x + ao, a, # 0. (a) Find (d"…

A: Please see the answer below.

Q: Let T„(x) be the Maclaurin polynomial of a function f(x) satisfying f(5) (x) < 1 for all x. Which of…

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Q: Suppose f(z) = u(x, y) is a real valued function on an open set U which we are viewing as a…

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Q: 3. (a) Find a unit in Z[√2] which is not equal to 1 or -1. Hint: Use 1 (b). (b) (*) Show that Z[√2]…

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Q: . There is a function f that's continuous at 0 and f(x) = f(2x) for all x's please show that f must…

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Q: 8. Let f be an entire function such that f(z)| < A + B|z|" for all z E C, where A and B are positive…

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Q: The minimum polynomial of 5'3 over Q is x 1/3

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Q: The Maclaurin polynomial of degree 4 for the function f(x) = cos(x) is %3D

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Q: Find a polynomial f with the following properties: i) f is concave up on (-, –2) U (0, ∞) and…

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Q: Show that f(x)= x14 – 6x + 75 is irreducible over Q.

A: Eisenstein's Irreducibility Criterion: Let us consider polynomial…