Problem 2 (20 points). Prove that a polynomial of odd degree has at least one real...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or 2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...
all odd degree polynomials have at least 1 real root. Explain how we can be sure. (the x-intercepts of a polynomial are the same as its real roots. Consider the “end behavior” of odd degree polynomials)
use the modulus maximum theorem to prove that every polynomial p(z) of degree > 1 has a root
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D_2p of a regular p-gon. Prove that f (x) has either all real roots or precisely one real root.
Prove that a tree with at least two vertices must have at least one vertex of odd degree.
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...
Problem 2 (2 points): Sketch a cubic function (third degree polynomial function) y x = 1 and x 4 and a loc p(x) with two distinct zeros at al maximum at x 4. Then determine a formula for your function. [Hint you will have one double root.] Sketch: Formula: p(x)-
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.
State the degree of the following polynomial equation. Find all of the real and imaginary roots of the equation, stating multiplicity when it is greater than one. X6 10x5 25x4-0 The degree of the polynomial is Zero is a root of multiplicity is a root of multiplicity 2.
Can someone please help me out State the degree of the following polynomial equation. Find all of the real and imaginary roots of the equation, stating multiplicity when it is greater than one The degree of the polynomial is Zero is a root of multiplicity is a root of multiplicity 2. Find a polynomial equation with real coefficients that has the given roots -1, 3,-4 The polynomial equation is x3--o Find a polynomial equation with real coefficients that has the...