2) Let f(T) e Q[T] be an irreducible polynomial of degree n > 1. Show that f(T) has n different roots in C.

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: Find a number a in [0, 1] such that f (a) = fave over the interval [0, 1]: f(x) = 3x² O الله O √3 O…

A: The average value of the function f(x) over the interval The given function is

Q: Let f (x) = x3 − e−x,                             x0 = 0.5,    x1 = 0.7,    x2 = 1.0. Find the…

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Q: the error estimate for Taylor polyno

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Q: Let f(x) = Q[x] be an irreducible polynomial and assume that f(x) splits in R. Let y E R be a root…

A: To prove that 2^(2/3) + gamma is a primitive element of Q(gamma, 2^(2/3))/Q, we need to show that it…

Q: c) Find the second degree polynomial approximation to f at (a, b) = (1, – 2) T2(x, y)

A: Given :- The second degree polynomial approximation to f at a,b = 1,-2 f x,y = -x2y2x2 +y2

Q: Let f be a function such that f(1.0) = 0.24, f(1.1) = 0.32, f(1.3) = 0.16, f(1.5) = 0.18. Use…

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Q: Determine if the following polynomials are irreducible or not k(1, y) = 1® – y° € Q!r, y]

A: Given:- k(x ,y)=x6-x7 ∈ Q x,y

Q: 2. Consider the Newton polynomial Pn(x) = ao+a1(xr-xo) + a₂(x-xo)(x − x₁) +... +an(x-xo)(x -…

A: Given that   that interpolate then we have Now  ,       putting

Q: In linear space, consider all true polynomial functions with internal multiplication x(t)y(t)dt for…

A: We will use GRAM-SCHMIDT ORTHOGONALIZATION PROCESS to solve this problem For a basis y0,y1,y2 to…

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 :

Q: - Let Q(x) be a polynomial of degree less than N. Prove that Q(x)PN(x)dx = 0.

A: Note, that this result is a specific case of orthogonality of Legendre’ s polynomials. We will prove…

Q: 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…

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