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7. Exercise 10.26. Suppose f is a polynomial function of degree n with f0 (so n must be even). Pr...
Please answer problem 4, thank you. 2. The polynomial p of degree n that interpolates a given function f at n+1 prescribed nodes is uniquely defined. Hence, there is a mapping f -> p. Denote this mapping by L and show that rl Show that L is linear; that is, 3. Prove that the algorithm for computing the coefficients ci in the Newton form of the interpolating polynomial involves n long operations (multiplications and divisions 4. Refer to Problem 2,...
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
Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every β E E can be expressed uniquely in the form β-bo-b10 + +b-1a-1 for some bi F. (a) Show every element can be written as f (a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ n_1α α0. Use this...
f(x) = x/(x+1), n = 4. We are asked to find the maclaurin polynomial of this function to the nth degree. Which means, we're supposed to differentiate the function,f(x) 4 times, then do something else(this is where im lost). Help, please.
Cs Saaet be the splitting field of a polynomial f of degree 5 over Q. Prove that E has no subfields F with [F : =7. Cs Saaet be the splitting field of a polynomial f of degree 5 over Q. Prove that E has no subfields F with [F : =7.
Explain why a polynomial function of even degree cannot be one-to-one?
approximate f by aTaylor polynomial with degree n at the number a and usetaylor's inequality to estimate the accuracy of theapproximation f(x)=when x lies in the given intervalf(x)=sin(x), , n=4,
For a), b), c), suppose P2(x) = a + bx + cx2is the second degree Taylor polynomial for the function f about x =0. What can you say about thesigns of a,b,c if f has the graphgiven below?a)b)c)
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.
(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)...