Question

(a) For a polynomial x^{2} + bx + c, give the
coefficients in terms of its roots: α1 and α2.

(b) For a monic, cubic polynomial, give the coefficients in terms of its roots.

(c) Generalize these result to monic polynomials of higher degree

Answer #1

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with
real coefficients.
1. Deduce that either f(x) factors in R[x] as the product of
three degree-one
polynomials, or f(x) factors in R[x] as the product of a
degree-one
polynomial and an irreducible degree-two polynomial.
2.Deduce that either f(x) has three real roots (counting
multiplicities) or
f(x) has one real root and two non-real (complex) roots that are
complex
conjugates of each other.

a)Give an example of a polynomial with integer coefficients of
degree at least 3 that has at least 3 terms that satisfies the
hypotheses of Eisenstein's Criterion, and is therefore
irreducible.
b)Give an example of a polynomial with degree 3 that has at
least 3 terms that does not satisfy the hypotheses of Eisenstein's
Criterion.

In the Problem 5 of Homework #1, for the given coefficients a=
1,b=−1e5 and c= 1, the root formula gives two real roots
r1= (−b−√∆)/(2a) = 1.0000003385357559e−05,
r2= (−b+√∆)/(2a) = 9.9999999989999997e+ 04
If we plug these two roots back into the quadratic function, we
obtain
f(r1) =−3.3844e−07,
f(r2) = 0
We see that root r1 is not correct up to the round-off
error.
(a) Explain what the problem is with the formula for
evaluatingr1numerically?
(b) Come up with a way...

Define T : P2 → R3 via T(a+bx+cx2) = (a+c,c,b−c), and let B =
{1,x,x2} and D ={(1, 0, 0), (0, 1, 0), (0, 0, 1)}.
(a) Find MDB(T) and show that it is invertible.
(b) Use the fact that MBD(T−1) = (MDB(T))−1 to find T−1. Hint: A
linear transformation is completely determined by its action on any
spanning set and hence on any basis.

(Sage Exploration) In class, we primarily have worked with the
field Q and its finite extensions. For each p ∈ Z primes, we can
also study the field Z/pZ , which I will also denote Fp, and its
finite extensions. Sage understands this field as GF(p).
(a) Define the polynomial ring S = F2[x].
(b) Find all degree 2 irreducible polynomials. How many are
there? For each,
completely describe the corresponding quadratic field extensions
of F2.
(c) True of false:...

What tools could AA leaders have used to increase their
awareness of internal and external issues?
???ALASKA AIRLINES: NAVIGATING CHANGE
In the autumn of 2007, Alaska Airlines executives adjourned at
the end of a long and stressful day in the
midst of a multi-day strategic planning session. Most headed
outside to relax, unwind and enjoy a bonfire
on the shore of Semiahmoo Spit, outside the meeting venue in
Blaine, a seaport town in northwest
Washington state.
Meanwhile, several members of...

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