Let f(x) = a_0 + a_1x + \cdots + a_nx^nf(x)=a 0 +a 1 x+⋯+a n x n be a polynomial of degree less than or equal to nn, and let \{x_0,x_1,\ldots,x_n\}{x 0 ,x 1 ,…,x n } be distinct points. What is the value of f[x_0,x_1,\ldots,x_n]f[x 0 ,x 1 ,…,x n ]? Explain.
f(x) is a polynomial in one variable x so it takes one value at a time. If x0, x1,.....xn are distinct points then f gives one value for each points which are f(x0), f(x1).....,f(xn). So f[x0,x1,.....xn] does not make sense.
Let f(x) = do +212 + ... + and" be a polynomial of degree less than or equal to n, and let {x0, 21, ..., Un} be distinct points. What is the value of f (x0, X1, ... , Xn]? Explain.
let fx be a polynomial of degree <= to n
whats the value of f(Xo, X1....Xn). explain
Let f(x) = ao tai xt...... + Anxh be a polynomial of | degree less than or equal to n, and let {xo.xi... n} be distinct points What is the value of f[xo, X.. Xn] Justify / Explain.
Let (r) 2 +ar+ ... +,r" be a polynomial of degree less than or equal to n and letto ......} be distinct points What is the value of from... Is? Explain Please select files) Select files)
Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
Please, please use MATLAB and
explain everything, thank you!!! If possible please plot the
values.
Consider the general problem of finding a root of f, that is, find a solution to the equation f(x) = 0. The secant method: y = p(x) equiv f(x_1) + (x - x_1) middot f(x_1) - f(x_0)/x_1 - x_0 Implement the secant method for solving the problem of finding a solution to the equation x = cos x. Use your computer implementation of the secant...
4. Another approximation for integrals is the Trapezoid Rule: integral (a to b)f(x) dx ≈ ∆x/2 (f(x_0) + 2f(x_1) + 2f(x_2) + · · · + 2f(x_n−2) + 2f(x_(n−1)) + f(x_n)) There is a built-in function trapz in the package scipy.integrate (refer to the Overview for importing and using this and the next command). (a) Compute the Trapezoid approximation using n = 100 subintervals. (b) Is the Trapezoid approximation equal to the average of the Left and Right Endpoint approximations?...
Question: Let f(x) be a function satisfying f(0) = 0, f'(0) = 5, f'(0) = -6 and |f(3)(x) = 6 for 0 5x51. Find the Taylor polynomial of degree 2 off at x = 0 and then find lim 5x-f(x) x2 x=0+ Answer: The Taylor polynomial of degree 2 off at x = 0 is P2(x) = Near x = 0, the function f(x) is equal to P2(x) plus some remainder, that is f(x) = P2(x) + R3(x).
Rings and fields- Abstract Algebra
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g() e Fx be any polynomial. Show that every irreducible factor of f(g()) E Flx] has degree divisible by n (b) (4 points) Prove that Q(2) is not a subfield of any cyclotomic field over Q.
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g()...
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
Question 4 (20 points) Let F: R R be any homogeneous polynomial function (with degree no less than one) with at least one positive value. Prove that the function f:Rn R, f(x) F(x) 1, defines on f-1(0) a structure of smooth manifold.