11. Prove that ifp is a polynomial of degree n , and ifp(a)-0, then p(z) =...
use the modulus maximum theorem to prove that every polynomial p(z) of degree > 1 has a root
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
p(z) =x2 (n-1) (z + 1 What is the degree of this polynomial function p(x)?
Let P(z) be a polynomial of degree n 2 1. Then CA. P(z) is analytic and bounded on C B. P(z) is not analytic but bounded on C C. P(z) is analytic and unbounded on C D. P(2) is bot analytic and unbounded on C
Let R(z)=Pn(z)/Qm(z), where Pn(z) is n-th polynomial and Qm(z)is m-th polynomial, prove the following statementm-n ≥ 2, Res[R(z), ∞] = 0.
Suppose that P is a polynomial of degree n and that P has n distinct real roots. Prove that P(k) has n-k distinct real roots for 1≤ k ≤ n-1.
Consider the polynomial P(n) of degree k: P(n) = aknk + ak-1nk-1+…..+ a1n + a0. with all ai > 0 Using the definition of Θ(nk), prove that P(n) € Θ(nk).
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
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,...
Here you are asked to prove the Fundamental Theorem of Algebra a different way by using Rouché's Theorem. Where n E N, consider the polynomial n-1 Pn (z)z" k-0 Using the circular contour C-[z : zR with R appropriately chosen, (a) prove that pn(2) has (counting multiplicity) precisely n zeros in the open disc D(0, R); (b) also show that Pn(z) has no zeros in C \ D(0, R) Here you are asked to prove the Fundamental Theorem of Algebra...