Polynomials Definition and 740 Threads

Homework Statement Let P denote the set of all polynomials whose degree is exactly 2. Is P a vector space? Justify your answer. Homework Equations (the numbers next to the a's are substripts P is defined as ---->A(0)+A(1)x+A(2)x^2 The Attempt at a Solution I really don't...

Homework Statement I am trying to work out a solution to the following problem, where we are working in a field K complete with respect to a discrete valuation, with valuation ring \mathcal{O} and residue field k. Q: Let f(X) be a monic irreducible polynomial in K[X]. Show that if f(0) \in...

Homework Statement Find x (17-x)^2(11-x)+256-32(17-x)-64(11-x)=0 Homework Equations The Attempt at a Solution This eq has 3 solutions. I solved this by multiplication. Is this some other easier way. Perhaps to group some of the factors 17-x and 11-x. Tnx for the answer.

Homework Statement Integrate the expression Pl and Pm are Legendre polynomials Homework Equations The Attempt at a Solution Suppose that solution is equal to zero.

I'm trying to prove the following, which is left unproven in something I'm reading on ruler-and-compass constructions: If ax^3+bx^2+cx+d is a polynomial over a subfield F of ℝ, and p+q\sqrt{r} is a root (with \sqrt{r}\notin F) then p-q\sqrt{r} is also a root. The theorem immediately before...

Given polynomials of degree n > 2, such that they have the form of p(x) \ = \ x^n \ + \ a_1x^{n - 1} \ + \ a_2x^{n - 2} \ + \ a_3x^{n - 3} \ + \ ... \ + \ a_{n - 2}x^2 \ + \ a_{n - 1}x \ + \ a_n. And \ \ all \ \ of \ \ the \ \ a_i \ \ are \ nonzero \ integers \ (which...

I know this may be a very stupid question, but I would really like to know. Is the determinant and the characteristic polynomial of an equation unique? I did several textbook questions and when I look at the solutions, they end up with completely different answers. Sometimes I am wrong and see...

http://bildr.no/view/1030479 The link above, it is my own and it is a bit disorderly, I think should explain taylor polynomials. In one assignent one had an assignment to derive taylor polynomials for cost^2 If one use the derivation rules with chain one get 2t for first derivative and...

we are given B = CAC^-1 Prove that A and B have the same characteristic polynomial given a hint: explain why ƛIn = CƛInC^-1 what I did was: B = CAC^-1 BC = CA Det(BC) = Det(CA) Det(B) Det(C) = Det(C) Det(A) Now they’re just numbers so I divide both sides by Det(C) Det(B) = Det(A)...

Hello everyone, Sorry if this is in the wrong sub-forum, I wasn't sure exactly where to place it. I was wondering if there is an orthogonality relationship for the Legendre polynomials P^{0}_{n}(x) that have been converted to cylindrical coordinates from spherical coordinates, similar to...

Homework Statement Nullity(B-5I)=2 and Nullity(B-5I)^2=5 Characteristic poly is: (λ-5)^12 Find the possible jordan forms of B and the minimal polynomials for each of these JFs. The Attempt at a Solution JFs: Jn1(5) or ... or Jni(5). Not sure how to find these jordan forms and minimal polynomials.

Does this actually work well? We won't learn isomorphisms in linear algebra, but a friend of mine showed me an example as I prefer to work with vectors and matrices rather than polynomials (All of my problem sets are with matrices and vectors). For example, if I wanted to find a basis for P3...

Homework Statement Let V be a finite dimensional complex vector space and T be the linear operator of V. Prove that the following are equivalent a V has a basis consisting of eigenvectors of T. b T can be represented by a diagonal matrix. c all the eigenvalues of T have multiplicity...

Homework Statement I'm given that the function f(x) is n times differentiable over an interval I and that there exists a polynomial Q(x) of degree less than or equal to n s.t. \left|f(x) - Q(x)\right| \leq K\left|x - a\right|^{n+1} for a constant K and for a \in I I am to show that Q(x)...

Is it true that polynomials of the form : f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a where \gcd(n+1,k+1)=1 , a\in \mathbb{Z^{+}} , a is odd number , a>1, and a_1\neq 1 are irreducible over the ring of integers \mathbb{Z}...

Homework Statement Do the polynomials t^{3} + 2t + 1,t^{2} - t + 2, t^{3} +2, -t^{3} + t^{2} - 5t + 2 span P_{3}? Homework Equations N/A The Attempt at a Solution My attempt: let at^{3} + bt^{2} + ct + d be an arbitrary vector in P_{3}, then: c_{1}(t^{3} + 2t + 1) + c_{2}(t^{2} -...

I have completed a difference table for 4 points, x0, x1, x2, x3 and found the third degree poly that goes through these four points. Now I need to know how to make the polynomial of second degree that interpolates x0, x2, and x3. Do I just need to remake the table for 3 points, now excluding...

Homework Statement Here is the entire problem set, but (obviously) you don't have to do it all, if you could just give me a few hints on where to even start, because I am completely lost. Recall that we found the solutions of the Schrodinger equations (x^2 - \partial_x ^2) V_n(x) =...

Hello all I had a simple question that I am intuitively sure I know the answer to but can't quite prove it. Suppose k is a polynomial in x and y, and k(x-1) = q for q some polynomial in y. Then is k = 0 ? How do I verify that k must be equal to 0? I can see that to just get a polynomial...

Let V= \mathbb{R}_3[x] be the vector space of polynomials with real coefficients with degree at most 3 and let D:V\to V be the linear operator of taking derivatives, D(f)=f'. I'm trying to check the Rank-nullity theorem for this example but it doesn't seem to hold: Since D is not injective...

Homework Statement Given the matrix: 0 1 0 0 0 1 12 8 -1 (sorry I don't know how to put proper matrix format) a) find polynomials a(λ)(λ+2)2+b(λ)(λ-3) = 1 (where a(λ) and b(λ) are the polynomials) The Attempt at a Solution Well the characteristic...

[SIZE="3"]help withe this tow Question please ? Q1: A pair of dice, one red and the other green, is rolled six times. We know that the ordered pairs (1, 1), (1, 5), (2, 4), (3, 6), (4, 2), (4, 4), (5, 1), and (5, 5) did not come up. What is the probability that every value came up...

I'm now studying the application of legendre polynomials to numerical integration in the so called gaussian quadrature. There one exploits the fact that an orthogonal polynomial of degree n is orthogonal to all other polynomials of degree less than n with respect to some weight function. For...

Hello all! I am trying to work through and understand the derivation of the Legendre Polynomials from Jackson's Classical electrodynamics. I have reached a part that I cannot get through however. Jackson starts with the following orthogonality statement and jumps (as it seems) in his proof...

Homework Statement Are the following statements true or false? Explain your answers carefully, giving all necessary working. (1) p_{1}(t) = 3 + t^{2} and p_{2}(t) = -1 +5t +7t^{2} form a basis for P_{2} (2) p_{1}(t) = 1 + 2t + t^{2}, p_{2}(t) = -1 + t^{2} and p_{3}(t) = 7 + 5t -6t^{2}...

Homework Statement Two Questions: 1. Prove, by contradiction, that if a and b are integers and b is odd,, then -1 is not a root of f(x)= ax^2+bx+a. 2. Prove, by contradiction, that there are infinitely many primes as follows. Assume that there only finite primes. Let P be the largest...

Homework Statement For p(x)=x4-2x3+3x2-3x+1 and A= 1 1 1 -1 -1 0 -2 1 0 0 1 0 1 0 0 0 you can check that P(A)=0 using this find a polynomial q(x) so that q(A)=A-1. The point is A4-2A3+3A2-3A=A(-A3+2A2-3A+3I)=I a) What is q(x)? I don't really understand how to approach...

Homework Statement Thanks very much for reading. I actually have two problems, I hope it's ok to state both of the in the same thread. 1. Let Vn be the space of all functions having the n'th derivitve in the point x0. I've been given the semi-norm (holds all the norm axioms other than ||v|| =...

Please I need Visual Basic algorithm for computing Hermite polynomials. Any one with useful info? Thanks.

Homework Statement Let f(x) = 2x^3 + 5x + 3 Find the inverse at f^-1(x) = 1 Homework Equations N/AThe Attempt at a Solution The only way that I know how to solve inverses is by solving for X, then replacing it by Y. Then I supposed I would sub 1 into the inverted polynomial. However I'm...

Homework Statement determine whether the following polynomials are irreducible over Q, i)f(x) = x^5+25x^4+15x^2+20 ii)f(x) = x^3+2x^2+3x+5 iii)f(x) = x^3+4x^2+3x+2 iv)f(x) = x^4+x^3+x^2+x+1 Homework Equations The Attempt at a Solution By eisensteins criterion let...

I am trying to find a way to integrate the following expression Integral {Ylm(theta, phi) Conjugate (Yl'm'(theta, phi) LegendrePolynomial(n, Cos[theta])} dtheta dphi for definite values of l,m,n,l',m' . You normally do this in Mathematica very easily. But it happens that I need to use this...

How are Hermite Polynomials related to the solutions to the Schrodinger equation for a harmonic oscillator? Are they the solutions themselves, or are the solutions to the equation the product of a Hermite polynomial and an exponential function? Thanks!

Hi, If I'm given something like this for a problem, Approximate the given quantities using Taylor polynomials with n=3 sqrt(101) how do I know what I should set f(x) equal to? I could set it to many different things, sqrt(x), sqrt(x+100), sqrt(x+50). My answer would be very different...

Hi! Im trying to do some rather easy QM-calculations in Fortran. To do that i need a routine that calculates the generalized Laguerre polynomials. I just did the simplest implementation of the equation: L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!} I implemented this in the...

I've been doing some work and I keep running into polynomials of the following form: P(x,y,z) = ax^2 + by^2 + cz^2 + 2(exy + fxz + gyz) \mod p where a,b,c \in \mathbb{Z}_p/ \{0\} and d , e, f \in \mathbb{Z}_p . It would be great if I knew anything about the existence of roots of P ...

I've just read a paper that references the use of student-t orthogonal polynomials. I understand how the Gauss-Hermite polynomials are derived, however applying the same process to the weight function (1 + t^2/v)^-(v+1)/2 I can't quite get an answer that looks anything like a polynomial...

Hi Could someone please explain how to best handle rational polynomials in Maple? I have matrix of rational polynomials and for some reason Maple keeps grumbling i.e. "error, (in, linearalgebra:- HermiteForm) expecting a matrix of rational polynomials" The matrix I am working with is...

while attempting to solve for radius of a circular orbit, I ended up getting P(x)=0 and dP(x)/dx=0 where P is a fourth order polynomial. I am not sure how can I solve it. Can someone shed some light on it. Thanks

Hi, yet another question regarding polynomials :). Just curious about this. Let f(x), g(x) be irreducible polynomials over the finite field GF(q) with coprime degrees n, m resp. Let \alpha , \beta be roots of f(x), g(x) resp. Then the roots of f(x), g(x), are \alpha^{q^i}, 0\leq i \leq n-1...

Hi, I'm currently doing a project and this topic has come up. Are there any known famous classes of polynomials (besides cyclotomic polynomials) that fit that description? In particular, I'm more interested in the case where the polynomials have odd degree. I know for example that the roots of...

hey i want to find out if the set s = {t2-2t , t3+8 , t3-t2 , t2-4} spans P3 For vectors, i would setup a matrix (v1 v2 v3 v4 .. vn | x) where x is a column vector (x , y ,z .. etc) and reduce the system. If a solution exists then the vectors span the space, if there are no solutions then...

given a set of orthogonal polynomials \int_{-\infty}^{\infty}dx P_{m} (x) P_{n} (x) w(x) = \delta _{m,n} the measure is EVEN and positive, so all the polynomials will be even or odd my question is if we suppose that for n-->oo \frac{ P_{2n} (x)}{P_{2n}(0)}= f(x) for a known...

Hi all, I am currently a 2nd year mathematics and physics student. I am working, for the first time, on my own research and just sort of getting my feet wet. I got in touch with a professor that studies Special Functions and he led me to the Legendre functions and associated Legendre...

Hi I was wondering since i have problems factoring any polynomial past 2nd degree i was wondering if anyone can show a way i can remember for finals ^_^. IE. let's say we have a 3rd degree polynomial. X^3 - 3X^2 +4 i tried looking it up but most don't show how they did the work so i can...

[-1]int[1]P(x)Q(x)dx P,Q\inS verify that this is an inner product.

Homework Statement There is a recursion relation between the Legendre polynomial. To see this, show that the polynomial x p_k is orthogonal to all the polynomials of degree less than or equal k-2. Homework Equations <p,q>=0 if and only if p and q are orthogonal. The Attempt at a...

1. Let \mathbb{R}[x]_n^+ and } \mathbb{R}[x]_n^- denote the vector subspaces of even and odd polynomials in \mathbb{R}[x]_n Show \mathbb{R}[x]_n=\mathbb{R}[x]_n^+ \oplus\mathbb{R}[x]_n^- 3. For every p^+(x) \in \mathbb{R}[x]_n^+ \displaystyle p^+(x)=\sum_{m=0}^n a_m x^m=p^+(-x)...

Let T:V to V be a linear operator on an n-dimensional vector space V. Let T have n distinct eigenvalues. Prove that the minimal polynomial and the characteristic polynomial are identical up to a factor of +/- 1 I'm probably over thinking this, but it seems that if you have n distinct...

In how many ways can obtain polynomial from [PLAIN]http://im3.gulfup.com/2011-05-05/1304543619801.gif notes that c any coffieceints is in{0.1} also in how many ways can obtain even ploynomials?whats the probability that we can obtain P(1,1,1)=0