Prove that every polynomial having real coefficients and odd degree has a real root

Problem: Prove that every polynomial having real coefficients and odd degree has a real root

This is a problem from a chapter 5.4 'applications of connectedness' in a book 'Principles of Topology(by Croom)'

So you should prove by using the connectedness concept in Topology, maybe.

Solutions

Expert Solution

milcah answered 1 year ago

Next > < Previous

Related Solutions

Find a polynomial function P of the lowest possible degree, having real coefficients, a leading coefficient...

Find a polynomial function P of the lowest possible degree, having real coefficients, a leading coefficient of 1, and with the given zeros. 1+3i, -1, and 2?

Does every polynomial equation have at least one real root? a. Why must every polynomial equation...

Does every polynomial equation have at least one real root? a. Why must every polynomial equation of degree 3 have at least one real root? b. Provide an example of a polynomial of degree 3 with three real roots. How did you find this? c. Provide an example of a polynomial of degree 3 with only one real root. How did you find this?

Prove: Every root field over F is the root field of some irreducible polynomial over F....

Prove: Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

1] Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are...

1] Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 2 and 5i are zeros; f(1) = 52 f(x)= ? (Type an expression using x as the variable. Simplify your answer.) 2] Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it...

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using...

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n= 4; -1,4, and 4+2i are zeros f(1)=-156

If p(z) is a polynomial of degree n and that if α is a root of...

If p(z) is a polynomial of degree n and that if α is a root of p(z) = 0, then p(z) factors as p(z) = (z−α)q(z) where q(z) has degree (n − 1). Use this and induction to show that a polynomial of degree n has at most n roots.

How to find the polynomial function with real coefficients, degree 5, zeros 1+i, -3, and 5,...

How to find the polynomial function with real coefficients, degree 5, zeros 1+i, -3, and 5, and P(0)=30 and P(4)= -70?

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo...

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo p2 . (Hint: Use that if a, b have orders n, m, with gcd(n, m) = 1, then ab has order nm.) (b) Prove that for any n, there is a primitive root modulo pn. (c) Explicitly find a primitive root modulo 125. Please do all parts. Thank you in advance

Prove that every natural number is odd or even.

Prove that the algorithm for computing the coefficients in the Newton form of the interpolating polynomial...

Prove that the algorithm for computing the coefficients in the Newton form of the interpolating polynomial involves n^2 long operations (multiplication and division).

Subjects