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Let P(z) be a polynomial of degree n 2 1. Then CA. P(z) is analytic and...
p(z) =x2 (n-1) (z + 1 What is the degree of this polynomial function p(x)?
11. Prove that ifp is a polynomial of degree n , and ifp(a)-0, then p(z) = (z-a)q(z), where q is a polynomial of degree
Problem 2 Let f(x) = sin 2x and P() be the interpolation polynomial off with degree n at 20,***, Im Show that \,f(z) – P() Sin+1 – 20) (1 - 11). (I – In).
(proof) n all 26. Let P(z) = 0 stand for an the zeros of which are in the unit circle |z| < 1. Replacing each coefficient of P() by its conjugate we obtain the polynomial P(2). We define p*()=P( The roots of the equation P(z) + P*(2) = 0 are all on the unit circle |z| = 1 algebraic equation of degree n all 26. Let P(z) = 0 stand for an the zeros of which are in the unit...
2. (a) Let P, =Span{1, x, x?, x°, x*} be the collection of polynomial with degree at most 4. Con- sider subspace H = Span{1,x, x*}. Prove that H is a subspace of Pg. Find a basis for the subspace H. (b) Now consider the differential operator D : H P, defined by D(1) = 0, D(x) = 1 and D(x3)=3x2. Why this defined linear operator D:H-P? Is the map Donto? Is the map Done-to-one?
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
Turing machines, closure properties, decidability Let P(z) = coz" + ca"-! + . . . +4-12+ cn be a polynomial with a root at r = zo. Let Cmax be the largest absolute value of a . Show that 1. (a) rol S col (b) What can you conclude from (a) about the problem of deciding whether a given poly- (c) What can you conclude from (a) about the problem of deciding whether a given poly- (d) Apply the knowledge...
3. Let Tn(x) be the degree n Chebyshev polynomial. Evaluate Tn (0.5) for 2 <n < 10, by applying the three-term recurrence directly with x = 0.5, starting with T.(0.5) = 1 and Ti(0.5) = 0.5.
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...