Let f(x) = do +212 + ... + and" be a polynomial of degree less than...
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) = 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.
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
19. Use Newton's divided difference formula to find the polynomial of degree less than or equal to four, that cos 2, at the interpolation points 0. π/2. π. 3π/2. 2π. Do not approximate π by a number interpolates f(x) with finite digits.
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Problem 5 Let p.) be a polynomial satisfying the same constraints as in the previous problem and let (2) be given as in the preceding problem. Show that p.) = r(c)(c) for some polynomial r(c). Hint: you can use the fundamental theorem of linear algebra and the generalized product rule for derivatives Problem 4 Prove that the polynomial q(x) given by g(x) = II (2 – x;) satisfies the linear constraints 9(wo) = 0, d'(x0) = 0, ......
let k be a field. 4. Conclude that a factorization of a polynomial f(x) of positive degree as g(x)h(x) is nontrivial iff the factors g(x) and h(x) have degrees strictly less than the degree of f(x).
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.
Question 1 2 pts The Hermite Interpolation polynomial for 33 distinct nodes has Degree at most {Be Careful with the answer. Look at the Theorem and the Question Carefully; compare the given information} Question 2 2 pts If f € C4 [a, b] and p1, P2, P3, and p4 are Distinct Points in [a, b], Then 1. There are two 3rd divided differences 2. There is exactly one 3rd divided difference and it is equal to the value of f(iv)...
Let f : [x0−h, x0+h]→R be defined. (a) Construct the 2nd degree Lagrange polynomial fitting {x −h,x ,x +h} and compute P′′(x ). (b) Use Taylor’s theorem to derive the same formula with error term.