Intermediate Value Theorem for Polynomials

Rhythmer
Messages
14
Reaction score
0
Prove: if P is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0
 
Physics news on Phys.org
You have the IVT, so what's the problem?
 
matt grime said:
You have the IVT, so what's the problem?
I need to prove the IVT. Our professor told us there is such proof but it's not an easy one.
 
If you insert some words like;

proof of the intermediate value theorem

into google you get lots of proofs. All you need to do is justify that polynomials are continuous, and that is easy. Also this has nothing to do with number theory. It is analysis/calculus.
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
Replies
48
Views
3K
I Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
Replies
0
Views
2K
I Finite fields, irreducible polynomial and minimal polynomial theorem
Replies
6
Views
3K
MHB Orthogonal Complement of Polynomial Subspace?
Replies
2
Views
2K
I Analogy question for algebraists
Replies
1
Views
2K
I Splitting ring of polynomials - why is this result unfindable?
Replies
9
Views
2K
I Equivalent definitons for primitive polynomials
Replies
24
Views
357
I Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices
Replies
14
Views
3K
I What can be deduced about the roots of this polynomial?
Replies
3
Views
3K
I Matrix rank in terms of determinant polynomial
Replies
14
Views
3K

Hot Threads

Recent Insights

Back
Top