Applications of the Mean Value Theorem: Roots of Polynomials

Show that if the roots of the polynomial p are all real, then the roots of p' are all real. If, in addition, the roots of p are all simple, then the roots of p' are all simple.

Solution Preview

The solution:

Let the roots of the polynomial equation p(x)=0 are real. Let a and b are any two roots of p(x). So, We have the following conditions satisfied by p(x)

1. p(x) is continuous as it is a polynomial
2. p(x) is differentiable for the same reason
3. p(a)=p(b)

Hence ...

Solution Summary

Real and Simple Roots of Polynomials are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Show that there are exactly (p^2-p)/2 monic irreducible polynomials of degree 2 over Z_p, where p is any prime.
Using the definition of irreducibility, Theorem: A polynomial of degree 2 or 3 is irreducible over the field F iff it has no roots in F, or Lemma of Theorem: The nonconstant polynomial p(x) an element of F[x] is irr

Determine whether thepolynomials have multiple roots.
See attached file for full problem description.
19. Let F be a field and let f(x) =...... The derivative, D(f(x)), of f(x) is defined by
D(f(x)) = ......
where, as usual, ....... (n times). Note that D(f(x)) is again a polynomial with coefficients in F.
The polynomi

Use Rouche's Theorem to determine the number of zeros, counting multiplicities, of thepolynomials inside the given regions.
a) z^4 + 3z^3 + 6 inside the circle |z| = 2
b) 2z^5 - 6z^2 + z + 1 in the annulus 1 <= |z| < 2
Thanks

2. verify that the function f(x) = e^-2x satisfies the hypothesis of meanvaluetheorem on the interval [0, 3] and find all the number c that satisfy the conclusion of themeanvaluetheorem.
3. show that a polynomial of degree two has at most two real roots.

This question is about the Weierstrass Approximation Theorem
Show that the Approximation Theorem does not hold if we replace I by R(real number system), by showing that if f(x) = e^x for all x, then f:R->R cannot be uniformly approximated by polynomials.

Please see the attached file for the fully formatted problems.
---
- Suppose the function f is analytic inside...
- Determine the number of zeros...
- Write f(z) - z^n and...
- Any polynomial...
---
(See attached file for full problem description)

I really need some help with this. Someone has got to have the background for this one.
"Let p1(x) = x^3 + x + 1 and p2(x) = x^3 + s + 2 in F5[x]. F5[x] is ust the set of all polynomials in x with coefficients from the set {0,1,2,3,4} with arithmetic done mod 5. Compute (x+2)^2114 in F5[x]/(p1(x)) and in F5[x]/(p2(x)). The

Find all the complex cube roots of w= 8(cos 150 + i sin 150). Write theroots in polar form with theta in degrees.
answers look like;
z sub 0 = ? (cos ?degrees + i sin ?degrees)
z sub 1 = ? ( cos ?degrees + i sin ?degrees)
z sub 2 + ? ( cos ?degrees + i sin ?degrees)
First build the expression for x sub k with r=8, th