Prove that, for each positive n, a polynomial of degree n has at
most n roots.
Please show the solution with each steps clearly.
Thank you.
Prove that, for each positive n, a polynomial of degree n has at most n roots....
with distinct nodes, prove there is at most one polynomial of
degree ≤ 2n + 1 that interpolates the data. Remember the
Fundamental Theorem of Algebra says a nonzero polynomial has number
of roots ≤ its degree. Also, Generalized Rolle’s Theorem says if r0
≤ r1 ≤ . . . ≤ rm are roots of g ∈ C m[r0, rm], then there exists ξ
∈ (r0, rm) such that g (m) (ξ) = 0.
1. (25 pts) Given the table...
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.
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,...
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
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
(c) Iff is a polynomial function of degree n, then f has, at most, n-1 turning points. First, identify the degree of f(x). To do so, expand the polynomial to write it in the form f(x) = a,x"+an-1*"-1 + ... + a,x+20- f(x) = + (2x2 + 7)? (x2+6)
Find a third-degree polynomial equation with rational coefficients that has roots -4 and 2 + i.
The polynomial of degree 4
The polynomial of degree 4, P(x) has a root of multiplicity 2 at x = 4 and roots of multiplicity 1 at x = 0 and x = – 2. It goes through the point (5, 7). Find a formula for P(x). P(x) =
The polynomial of degree 5, P(2) has leading coefficient 1, has roots of multiplicity 2 at I = 1 and I = 0, and a root of multiplicity 1 at I = - 3 Find a possible formula for P(x). P(x) = Question Help: Video Submit Question
Using the complex-n-th roots theorem:
5. (a) Use Theorem 10.5.1: Complex n-th Roots Theorem (CNRT) to com- pute all the 4-th roots of -1/4. (b) Factor the polynomial 4x4 + 1 in C[x]. (c) Factor the polynomial 4x4 +1 in R[x]. (d) Use Rational Roots Theorem to prove that the polynomial 4x4 + 1 has no rational roots. Deduce the factorization of 4x4 + 1 in Q[x].