1. Let f(3) = 20.3" +ajx"-1 + ... + an, where ao, ar- an are integers....
1.(1) Let A={f(x): f(x)-axx? +ajx + ap} where a, eR (i=1,2,3). Define f+g by (f+g)(x)=(a+b)x² + (a1 +b ) x + (ao+b) also define (rf)(x)=(ra) x? +(ra)x+rao Show that A is vector space.
2. Let 6 marks (a) Find f(x),f"(x), and f"(x). (b) Find the second order Taylor expansion of f at 1, namely f(r) = ao + ala-1 ) + a2(z-1)2 + R2(x), where Ra is the remainder. You should find ao, a, a2, and R(p). 8 marks that the error in this estimation (i.e., R2(0.9)1) is at most 10-3. 6 marks (c) Use the Taylor expansion found above to estimate the value of f(0.9). Show Find f(x), f"(), and f" (b)...
1 #6: Let T: P2 → p2 be defined by T(ao +ajx + a2 x2) = (Tao + 381 +8a2) – (a1 + 36a2)x+ 20 x2 Find the eigenvalues of T. Enter any repeated eigenvalues as often as they repeat. em #6:
Show that every positive integer n, there is a string of n consecutive integers where first integer is even, the second is divisible by a perfect square(other than 1), the third by a perfect cube(other than 1), etc..., and the nth is divisible by the nth power of an integer(other than 1). Then find an example for n = 3.
Problem #6: Let T:p2 → p2 be defined by T(ao +ajx + a2 x2) = (890 +6a1 + 902) – (a1 + 36a2)x + (20 – 4a2) x2 + Find the eigenvalues of T. Enter any repeated eigenvalues as often as they repeat. Problem #6: Just Save Submit Problem #6 for Grading Problem #6 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark:
Problem #6: Let T:p2 → 2 be defined by T(ao +ajx + a2 x2) = (7a0 + a1 - 7a2) - (a1 + 25a2)x+ 20 x2 Find the eigenvalues of T. Enter any repeated eigenvalues as often as they repeat. Problem #6: Just Save Submit Problem #6 for Grading Attempt #1 Attempt #2 Attempt #3 Problem #6 Your Answer: Your Mark:
(3). Let F be a field and let f(x) = ao-chx +-.. + an-,Kn-1 + an&n E F[x]. Prove that x - 1 is a factor of f(x) if and only if ao+ aan 0
Theorem. Let p(x) = anr" + … + ao be a polynomial with integer coefficients, i, e. each ai E Z. If r/s is a rational root of p (expressed in lowest terms so that r, s are relatively prime), then s divides an and r divides ao Use the rational root test to solve the following: + ao is a monic (i.e. has leading coefficient 1) polynomial with integer coefficients, then every rational root is in fact an integer....
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
14. Let ao = 1 and let am+1 = 2am + 1 for all positive integers m 0, Find an explicit formula for am (in terms of m only) and prove your formula is correct.