(b) Let f be a Cn function on an open interval [a, b]. Assume that the set of zeros {x € (a, b) : f(x) = 0} has at least (n + 1) elements. Prove that there exists xo E (a, b), such that f(m) (xo) = 0. Use (b) to prove that a polynomial (with real coefficients) of degree n has at most n real roots.

Q: Let f(x) be a polynomial function whose coefficients areintegers. Suppose that r is a real zero of f…

A: Consider an example:  2x3=x+6x+12x-42x3=x3+14x2-2882x3-x3-14x2+288=0⇒x3-14x2+288=0

Q: Let T(x) denote the Chebyshev polynomial of degree k. Express the composite polynomial Tn(Tm(x)) of…

A: The Chebyshev polynomial of first kind defined as  Tmx=2xTm-1x-Tm-2x, with initial conditions given…

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

A:

Q: Let F = Q, s(x) = x² + 1, and E = Q[r]/(x² + 1)Q[x]. i) Factor the polynomial r? +1€ Q[r] into…

A:

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: Yazı Tipi Paragraf Stiller Let f(x) = x/3. By using quadratic Lagrange interpolating polynomial for…

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…

A:

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

A:

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: According to the integral zero theorem, the possible integral zeros of f(x) = 4x³+5x²+7x+6 are…

A:

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: 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: 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: let f(z) be polynomial of degree at least two. show that |f(z)|>|z| for large |Z|.

A: We prove this result by contradiction.

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<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)

A:

Q: Let f(r) be a polynomial function such that f"(x) = xª – 5z + 6z². Find the æ values for the…

A:

Q: Let f(z) be the entire (analytic everywhere) function, such that |f(z)|<C|z|^N for large z, i.e. the…

A:

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| &gt; 100} f satisfies f(Z)&lt;a+ b |z|^n where a ,b are…

Q: .3 Let f(x) = Use the Bisection method three times to find three successively smaller intervals…

A: Given: f(x)=-x3+2x-2 Here, f(-4)=54 and f(0)=-2 So, The given function is continuous, and the root…

Q: Define = -ĤI ( ¹ - -¦ ) = (¹ - ¹) · (¹ – ¦ ) · (¹ - -) --- (¹ ---), 1=2 where ne Z+ and n > 2. Here,…

A:

Q: Let p: R → R be a polynomial of odd degrees. Prove that the equation p(x) = 0 has a solution in real…

A: Given p:R→R be a polynomial of odd degree. We have to prove that the equation px=0 has a solution in…

Q: Find the (natural) domain of the function x + 1 2х — 1 f (x) = Select one: a. (-0, –1] U (1/2, ∞) b.…

A:

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

A:

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

A:

Q: Let Q(x) be any polynomial of degree ≤ n with real coefficients and let M be the maximum of Q(x)| on…

A:

Q: 4. Consider the polynomial f(x) = x² + 1. (a) Show that f(x) is irreducible in (Z/3Z) [x]. (b) Show…

A:

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: 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…

A:

Q: Consider the function f(x) = x = [0,2]. Construct the , Lagrange interpolation polynomial which…

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 :

Q: Let f(x) be a function on (- 0, 0) and f(x + 2) = f(x - 2). If f(x) = 0 has only three real roots in…

A: Introduction :